As per Roman Catholics, Ministry of High Priest of Jesus is still continuing.
CCC 1410: It is Christ himself, the eternal high priest of the New Covenant who, acting through the ministry of the priests, offers the Eucharistic sacrifice. And it is the same Christ, really present under the species of bread and wine, who is the offering of the Eucharistic sacrifice.
Christ’s Sacrifice is once for all.
And by that will, we have been made holy through the sacrifice of the body of Jesus Christ once for all.
Catechism of Catholic Church teaches that this is how Israel understands its liberation from Egypt: Every time Passover is celebrated, the Exodus events are made present to the memory of believers so that they may conform their lives to them. This memorial is not merely the recollection of past events but the proclamation of the mighty works wrought by God for men.
In the New Testament, the memorial takes on new meaning. When the Church celebrates the Eucharist, she commemorates Christ's Passover, and it is made present the sacrifice Christ offered once for all on the cross remains ever present. In the liturgical celebration of these events, they become in a certain way present and real.
Because it is the memorial of Christ's Passover, the Eucharist is also a sacrifice. The sacrificial character of the Eucharist is manifested in the very words of institution: "This is my body which is given for you" and "This cup which is poured out for you is the New Covenant in my blood." In the Eucharist Christ gives us the very body which he gave up for us on the cross, the very blood which he "poured out for many for the forgiveness of sins. The Eucharist is thus a sacrifice because it re-presents (makes present) the sacrifice of the cross, because it is its memorial and because it applies its fruit:
[Christ], our Lord and God, was once and for all to offer himself to God the Father by his death on the altar of the cross, to accomplish there an everlasting redemption. But because his priesthood was not to end with his death, at the Last Supper "on the night when he was betrayed," [he wanted] to leave to his beloved spouse the Church a visible sacrifice (as the nature of man demands) by which the bloody sacrifice which he was to accomplish once for all on the cross would be re-presented, its memory perpetuated until the end of the world, and its salutary power be applied to the forgiveness of the sins we daily commit.
St. Augustine admirably summed up this doctrine that moves us to an ever more complete participation in our Redeemer's sacrifice which we celebrate in the Eucharist:
This wholly redeemed city, the assembly and society of the saints, is offered to God as a universal sacrifice by the high priest who in the form of a slave went so far as to offer himself for us in his Passion, to make us the Body of so great a head. . . . Such is the sacrifice of Christians: "we who are many are one Body in Christ" The Church continues to reproduce this sacrifice in the sacrament of the altar so well-known to believers wherein it is evident to them that in what she offers she herself is offered
It is by the conversion of the bread and wine into Christ's body and blood that Christ becomes present in this sacrament. The Church Fathers strongly affirmed the faith of the Church in the efficacy of the Word of Christ and of the action of the Holy Spirit to bring about this conversion. Thus St. John Chrysostom declares:
It is not man that causes the things offered to become the Body and Blood of Christ, but he who was crucified for us, Christ himself. The priest, in the role of Christ, pronounces these words, but their power and grace are God's. This is my body, he says. This word transforms the things offered.
And St. Ambrose says about this conversion:
Be convinced that this is not what nature has formed, but what the blessing has consecrated. The power of the blessing prevails over that of nature, because by the blessing nature itself is changed. . . . Could not Christ's word, which can make from nothing what did not exist, change existing things into what they were not before? It is no less a feat to give things their original nature than to change their nature.
As often as the sacrifice of the Cross by which 'Christ our Pasch has been sacrificed' is celebrated on the altar, the work of our redemption is carried out.
CCC 1348 All gather together. Christians come together in one place for the Eucharistic assembly. At its head is Christ himself, the principal agent of the Eucharist. He is high priest of the New Covenant; it is he himself who presides invisibly over every Eucharistic celebration. It is in representing him that the bishop or priest acting in the person of Christ the head (in persona Christi capitis) presides over the assembly, speaks after the readings, receives the offerings, and says the Eucharistic Prayer. All have their own active parts to play in the celebration, each in his own way: readers, those who bring up the offerings, those who give communion and the whole people whose "Amen" manifests their participation.