One time Jesus said: "I am Alpha and Omega, the beginning and the ending" (cf. Revelation 22:13). But at another time Jesus said that He is the everlasting God, which supposedly has no ending. Can anyone explain this seemingly contradictory statement?
The saying "I am the Alpha and Omega, the Beginning and the End" can be, and has been, interpreted in the following four ways at least:
(1) As an idiom meaning eternal. Thus Albert Barnes interprets it in Barnes' Notes. Adam Clarke follows the same interpretation in his commentary on Revelation 1:8 where he says:
I am Alpha and Omega - I am from eternity to eternity. This mode of speech is borrowed from the Jews, who express the whole compass of things by א (aleph) and ת (tau), the first and last letters of the Hebrew alphabet; but as St. John was writing in Greek, he accommodates the whole to the Greek alphabet, of which Α (alpha) and Ω (omega) are the first and last letters. With the rabbins מא ועד ת (meeleph vead tau), “from aleph to tau,” expressed the whole of a matter, from the beginning to the end. So in Yalcut Rubeni, fol. 17, 4: Adam transgressed the whole law from aleph to tau; i.e., from the beginning to the end. Ibid., fol. 48, 4: Abraham observed the law, from aleph to tau; i.e., he kept it entirely, from beginning to end. Ibid., fol. 128, 3: When the holy blessed God pronounced a blessing on the Israelites, he did it from aleph to tau; i.e., he did it perfectly.
So that the argument is that Alpha and Omega denote the whole of eternity.
Others have interpreted it in more limited senses:
(2) With respect to creation. That he is the Beginning, as in the Creator, and the End, as in the one who destroys the world at the end. This is how it is interpreted, for example, in the note on Rev 1:8 in an old commentary called The People's New Testament, or Johnson's Notes:
I am the Alpha and the Omega. The first and last letters of the Greek alphabet; hence "the beginning and the end." All begins with God and he closes the drama of earthly history.
The first two interpretations, obviously, would be the interpretations most acceptable to Trinitarians. The following interpretations are more likely to be accepted by non-Trinitarians.
(3) With respect to the church. That Jesus is to the church the beginning and end of everything, the beginning and end of salvation, etc. This is the sense in which the words were generally interpreted by the Socinians, as the following quotation from a note by an eminent Socinian (Wissowatius) in the Racovian Catechism (pages 161-162 of Thomas Rees' translation) will show:
That the Son of God is absolutely the First, none will venture to assert, who maintain that the Father is the first person of the Trinity. Erasmus well observes in his Annotations on John viii. 25: Quod in Apocalypsis dicitur principium et finis, constat intelligendum, Christum esse initium et consummationem Eccleslae, quam priore adventu constituit, posteriore perficiet. "As to what are called in the Revelation 'the beginning and the end,' it is evident that we must understand by them that Christ is the beginning and the consummation of the Church, which was founded by his first, and will be completed by his second, appearance." We read nearly to the same effect in Hermas Simil. 9; where, speaking of the Church, he calls the Son of God, "the old rock and the new gate;" and for these reasons, because the Church is founded upon him, and is older than every creature, and because he will in the last days appear for its completion, that those who are about to be saved may enter through him into the kingdom of God: which accords with the words of Paul, whose disciple he was (Col i. 15,18), and also with Revelation i. 17, and xxii. 13, as Grotius also has remarked. To the same effect are likewise Heb. ii. 10; xii. 2; Acts iii. 15, 31; Eph ii. 20, &c. B. WISSOWATIUS.
Sometimes in connection with this explanation, Hebrews 12:2 is used (as mentioned in the note quoted): "Looking unto Jesus the author and finisher of our faith..."
(4) With respect to an authority derived from the Father. An Arian interpretation is given in the following words in Scripture proofs and Scriptural illustrations of Unitarianism by John Wilson (page 201):
The terms, Alpha and Omega, beginning and end, are, in a finite sense, justly applicable to Jesus as the first of all created existences, and the last of those who will be required to resign the authority with which he is invested by the Father.--RAMMOHUN ROY on Rev. xxii. 13: Second Appeal, Appendix No. 2.
From page 200 of the same, what we could say is a fifth interpretation emerges, which is a much harder sell than all the others:
(5) That Jesus doesn't call himself Alpha and Omega. This argument is based around the fact that some manuscripts say "Lord God" in verse 8, not just "Lord," and some manuscripts omit the saying "I am the Alpha and Omega" in verse 11. One eminent textual critic of the late 1700s and early 1800s, Griesbach, presents the text of verse 8 with "Lord God" and verse 11 without "I am the Alpha and Omega" which gave the unitarians of that period an argument that only the Father says "I am the Alpha and Omega" and the Son does not. I checked that Griesbach really does present the text this way, and I checked the UBS 3rd Edition, and it presents these two verses the same way as Griesbach. The ESV, which follows a Nestle-Aland/UBS text thus reads:
Revelation 1:8 ESV “I am the Alpha and the Omega,” says the Lord God, “who is and who was and who is to come, the Almighty.” verse 9 I, John, your brother... verse 10 I was in the Spirit on the Lord's day, and I heard behind me a loud voice like a trumpet verse 11 saying, “Write what you see in a book and send it to the seven churches, to Ephesus and to Smyrna and to Pergamum and to Thyatira and to Sardis and to Philadelphia and to Laodicea.”
So unlike in the KJV, you find no statement here by Jesus in verse 11 that he is the Alpha and the Omega and the statement in verse 8 is presented not as from "the Lord" but as from "the Lord God." I don't know what anyone relying on that argument would do with Revelation 22:13, however, since that would seem to be a major problem for this theory (but I would assume that even without manuscript evidence they would view it as an interpolation or claim its the Father speaking here):
Revelation 22:12 ESV “Behold, I am coming soon, bringing my recompense with me, to repay each one for what he has done. verse 13 I am the Alpha and the Omega, the first and the last, the beginning and the end.”
So basically, every different Christological position that exists in Christianity can find a way to make these words fit their position, at least to their own satisfaction.
It is possible that Jesus is in this particular case was emphasizing what God said back in the Old Testament:
Isaiah 41:4 Who hath wrought and done it, calling the generations from the beginning? I the LORD, the first, and with the last; I am he.
Isaiah 44:6 Thus saith the LORD the King of Israel, and his redeemer the LORD of hosts; I am the first, and I am the last; and beside me there is no God.
Isaiah 48:12 Hearken unto me, O Jacob and Israel, my called; I am he; I am the first, I also am the last.
This was said by God to emphasize that he alone was God, He was before any other god was conceived and he will out last all of them.
But the most likely probability is that what he was referring to is:
1st. Corinthians 15:45 And so it is written, The first man Adam was made a living soul; the last Adam was made a quickening spirit.
Which is to summarize the fall and resurrection of mankind. Adam through his willful disobedience of God (Adam was not deceived he understood that he was disobeying God, yet did so of his own free will. Eve on the other hand was deceived and her disobedience was not totally willful), as a counter; to Adams willful disobedience, Jesus willful obedience brought an end to man's loss of acceptability. As Adam willfully accepted death as a punishment for his disobedience, Jesus willfully banned death as man's punishment by his obedience even unto a horrible death.
While I realize that you are seeking a more definitive answer there appears to be none in the Bible.
Hope this at least gives you some optional answers.
I can find a number of times in the New Testament (KJV) where the word "everlasting" is used, but they never refer to any person, not even the person of Jesus Christ, so without a citation (book, chapter, verse, translation), your question is based upon a false premise.
As to the phrse "alpha and omega, the beginning and the end", this is as much a poetic or literary device, rather than a description to be taken literally.