What significance did Rosh Hashanna have for early Christians?

Question

This question is asked on the day that the Jewish holiday Rosh Hashanna, the celebration of the new year, begins in 2024. The institution of the holiday is found in the Book of Leviticus, chapter 33:

23 And the Lord spoke to Moses, saying, 24 “Speak to the people of Israel, saying, In the seventh month, on the first day of the month, you shall observe a day of solemn rest, a memorial proclaimed with blast of trumpets, a holy convocation. 25 You shall not do any ordinary work, and you shall present a food offering to the Lord.”

The gospels depict Jesus as participating in several Jewish holidays but not this one, as far as I know. Based on the New Testament and the writings of the Church Fathers, does this holiday figure in early Christian tradition?

Answer 1

Based on the New Testament … does this holiday figure in early Christian tradition?

The Bible mentions the early Christians celebrating all three of the Spring festivals and most of the four Fall festivals, so it's almost certain that the early Christians also continued celebrating the Feast of Trumpets (Rosh Hashanah), the first of the Fall holidays.

Much of that festival's prophetic significance wasn't available until after John recorded the Book of Revelation, and most Christian writings and history for the next century or so seem to have disappeared, so we can't know how much of this holiday's significance the early Church understood.

But some churches today do have a strong tradition for what it means.
For instance, The Exciting Feast of Trumpets | United Church of God:

The Feast of Trumpets, one of the Holy Days of the Bible, pictures future times of both sadness and gladness—essential steps leading to the establishment of the Kingdom of God on earth!

A mighty trumpet announces the second coming of Christ

The Feast of Trumpets holds special excitement because it initiates the autumn festival season of four festivals. As with all the annual feasts, we need the New Testament to fully understand the meaning and symbolism.

Most interesting is what the Bible reveals about trumpet calls in connection with future prophesied events — events pointed to by the Feast of Trumpets!

Scripture tells us that an earth-shaking trumpet call will announce the most important and exciting event that will ever happen to humanity — the second coming of Jesus Christ! This will be the great climax and turning point of all history!

Note how Jesus described His future return: "Then the sign of the Son of Man will appear in heaven, and then all the tribes of the earth will mourn, and they will see the Son of Man coming on the clouds of heaven with power and great glory. And He will send His angels with a great sound of a trumpet, and they will gather together His elect from the four winds, from one end of heaven to the other" (Matthew 24:30-31).

Another great event: the resurrection of the saints

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The apostle Paul wrote: "For the Lord Himself will descend from heaven with a shout, with the voice of an archangel, and with the trumpet of God. And the dead in Christ [the saints who have died] will rise first. Then we [saints] who are alive and remain shall be caught up together with them in the clouds to meet the Lord in the air. And thus we shall always be with the Lord" (1 Thessalonians 4:16-17).
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Then Paul states when this resurrection will occur: "Behold, I tell you a mystery: We shall not all sleep [meaning not all in this resurrection will be dead in the grave when Christ returns], but we shall all be changed—in a moment, in the twinkling of an eye, at the last trumpet. For the trumpet will sound, and the dead will be raised incorruptible, and we shall be changed. For this corruptible must put on incorruption, and this mortal must put on immortality" (1 Corinthians 15:51-53).

A scroll sealed with seven seals

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The end-time sequence of events is pictured as Christ unrolling a scroll by breaking open its seven seals one by one (Revelation 5). The sequence of the seals is the same sequence of events Jesus described in His prophecy about His return recorded in Matthew 24, Mark 13 and Luke 21. These are crises that will be growing in frequency and intensity leading up to that time.

The opening of the first four seals unveils what are commonly referred to as the "four horsemen of the Apocalypse," representing false religion, war, famine and death by pestilence and various disasters (see Revelation 6:1-8).
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The seventh seal: the Day of the Lord

The heavenly signs of the sixth seal announce the Day of the Lord, with fearful people crying out: "For the great day of His wrath has come, and who is able to stand?" (Revelation 6:17). Likewise, the prophet Joel tells us that the heavenly signs will happen just "before the coming of the great and awesome day of the Lord" (Joel 2:31). So the period of the seventh seal that follows in Revelation is the Day of the Lord.

The opening of the "seventh seal," which is the Day of the Lord, reveals "seven angels" with "seven trumpets" (Revelation 8:1-2). Each trumpet blast announces terrifying calamities that will afflict rebellious mankind.
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The blowing of the seven trumpets

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When the seventh angel sounds, he is announcing events both wonderful and terrifying. The wonderful events are the coronation of Jesus Christ as King of Kings, His return to rule all nations and the resurrection of the saints—"at the last trumpet" (1 Corinthians 15:52). And the terrifying events are a series of seven last plagues, as we'll see.

Revelation 11:15-18 proclaims: "The seventh angel sounded his trumpet, and there were loud voices in heaven, which said: ‘The kingdom of the world has become the kingdom of our Lord and of his Messiah [or Christ], and he will reign for ever and ever.'
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The warning in the seventh trumpet

Note what follows in Revelation 15:1: "Then I saw another sign in heaven, great and marvelous: seven angels having the seven last plagues, for in them the wrath of God is complete."

Revelation 16 goes on to describe the seven last plagues. The first is a terrible affliction on people, the second is on the sea and its creatures, and the third is on all the rivers and springs of water. The fourth plague is on the sun, which creates scorching heat. The fifth plague is darkness and extremely painful sores.
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The sixth angel announces a vast army coming from the east led by Satan and his demons. They "go out to the kings of the earth and of the whole world, to gather them to the battle of that great day of God Almighty" (Revelation 16:12-14).
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The seventh angel announces incredible cataclysms!
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Then comes the Commander and His heavenly cavalry for the final battle!

The triumphant return of Jesus Christ

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Then we read of the awesome arrival and conquest by Jesus Christ and His armies! "Now I saw heaven opened, and behold, a white horse. And He who sat on him was called Faithful and True, and in righteousness He judges and makes war" (Revelation 19:11).
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The symbolism of the Feast of Trumpets

So what does the Feast of Trumpets picture? It portrays everything that happens from the angel blowing the first trumpet to Christ conquering His enemies at the time of His return. It warns of global catastrophes and war and yet, at the same time, celebrates the coming of Jesus Christ as King and the raising to life of His followers. This paves the way for peace on earth at last as Jesus sets up His Kingdom and begins to rule the world! The next three annual Feasts explain the rest of the story.
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This festival is a time of rejoicing before God, as it depicts a time when our Creator will dramatically intervene in world affairs for the good of mankind. It will be a time of joyful worship as we review what this day pictures in the plan of God.

This Holy Day stirs excited anticipation for the time when, as Zechariah 14:9 puts it, "the Lord shall be King over all the earth"!

Answer 2

I mean, September 1st is the ecclesiastical new year of the Orthodox Church. It is known as the Indiction. It does coincide with Rosh Hashanah which is the 1-2nd day of the month of Tishrei.

According to Holy Tradition, Christ entered the synagogue on September 1st when he began his earthly ministry. This fulfilled the writings of Prophet Isaiah. And is found in Luke 4:17-21.

Christ might not be directly referenced as celebrating the new year. As a faithful Jewish child and young adult who never sinned he would have respected and celebrated them with his family and those who he lived with. I am not certain we need a specific verse or passage that mentions each holiday... it is enough to know that He was without sin and fulfilled the law.

On that day a shofar which is a ram’s horn is blown. The Lord spoke to Moses, saying: ‘Speak to the children of Israel’, saying, ‘in the seventh month, on the first day of the month, you shall have a rest, a memorial of trumpets, a holy convocation for you (Leviticus 23: 23-24).’ The Hebrew month of Tishrei is the beginning of the civil calendar for the Jews. The ecclesiastical calendar, however, begins with the month of Nisan.

For Orthodox Christians, the civil calendar starts in January and the ecclesiastical calendar in September. For the Jews, however, although Tishrei marks the beginning of the civil calendar, the High Holidays (Rosh Hashanah, Yom Kippur and Sukkot) are all celebrated in this month. The month of Tishrei usually begins in September but in some years could begin as late as October 5th. It was in the month of Tishrei that the Jews entered the Promised Land. (GoArch blog, "Happy new year?" -John Athanasatos)

This variation on when exactly the new year starts is now more fixed on September 1st within the Orthodox Church based on the tradition of the hierarchs signing the Patriarchal and Synodal "Tome for the Indictus" which has been going on since Constantine the Great.

It is a tradition in Constantinople at the Phanar that the Ecumenical Patriarch along with the Holy Sacred Synod concelebrate the Divine Liturgy at the Cathedral of St George on September 1st. (Ibid)

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The church celebrates similar things on this day as those observed by the Jews. That is to say we celebrate the God's Creation.

Holy Spirit, one in essence with the Son, the eternal Word, with Him You create and sustain all things visible and invisible: Bless the beginning of this year; bring peace to Your Orthodox people, through the prayers of the Theotokos and all the saints (1st Theotokion of Festal Vespers).

Usually the Ikon of God breathing life into Adam or the Ikon of Adam naming the animals is venerated on this day.

And it is indeed a celebration within the Church. Generally in the "Thanksgiving" type where we give thanks for the abundance of the fruits of the earth and pray for the new harvest that is about to be sown. As well as giving thanks for the gift of Life.

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I hope that answers your question, the holiday has become a permanent feature of the yearly cycle of feast days and is still celebrated, it just it no longer called the same name.