The Lord's Prayer is explicitly from the New Testament. It's in there twice, actually. A short version in Luke 11:2–4 and a longer version in Matthew 6:9-13. In both cases, it's introduced by saying, "When you pray, say". Matthew 6 gives also detailed instructions for prayer more generally (do it in private, etc.). Jesus on the Cross quotes from Psalm 22 ("My God, my God, why have you foresaken me?") indicating that Psalms were still in use in the early Christian community. But as books like Acts make clear, and some of the Paulene letters, part of the new Christian community felt that all Jewish law still applied, so they almost certainly continued with most if not all of the Jewish prayers.

The (generally agreed) letters of Paul are the earliest extant Christian writings (c.45-60AD), and they contain many prayers, as well as teachings, greetings, and other literature.

Argualbly Galatians, 1 Corinthians and 1 Thessalonians are his first, and the prayer at the end of the Thessalonians has notable links with First century Jewish / Late Second temple language, and can reliably dated to the first half of the first century AD:

1 Thess3:11 Now may our God and Father himself and our Lord Jesus clear the way for us to come to you. 12 May the Lord make your love increase and overflow for each other and for everyone else, just as ours does for you. 13 May he strengthen your hearts so that you will be blameless and holy in the presence of our God and Father when our Lord Jesus comes with all his holy ones.

The other of real note is "Maranatha", or "Our Lord, Come! It is found at the end of 1 Corinthians, and is supposed to be from the Aramaic from the earliest Christian communities in Judea.

I would suggest you go to r/academicbiblical to ask this question. They can answer this question better than I.

Although they can answer the question better, I will attempt an answer to the question. I would suggest one of the oldest Christian prayers is the Priestly Blessing in Numbers 6:24-26. It predates Christianity and was first used by Jews, but later also used by Christians.

Here is the Revised Standard Version of this prayer

24 [May] the Lord bless you and keep you.

25 [May] the Lord make his face to shine upon you, and be gracious to you.

26 [May] the Lord lift up his countenance upon you, and give you peace.

(I have added the word [May] because I think it better translates the intent. Some may disagree.)

The antiquity of this phrase is demonstrated by its inclusion in the Hebrew Bible. Its antiquity is also demonstrated by it being on the two tiny silver scrolls found by archeologists at Ketef Hinnom

I apologize to the moderators for using Wikipedia references. However I have done more than just use Wikipedia references.