Paul claims to be an apostle by virtue of his vision/s of Jesus.
He does not claim to be an apostle by virtue of Jesus ordaining him to any priesthood.
He does not claim to be an apostle by virtue of Peter, another of the twelve, or anyone else ordaining him to any priesthood either, though he does want to point out that years after the fact James and Peter and John accepted him (Galatians 2:6-10).
When he refers to apostles (arguably including the woman Junia in Romans 16:7) and deacons (clearly including the woman Phebe in Romans 16:1-2), he does not have any priesthood or ordination to it in mind.
In the pastoral epistles attributed to Paul (but arguably not written by him), where the qualifications of a bishop are listed, no priesthood or ordination to it comes up. Ordination of elders comes up (they are ordained in the KJV of Titus 1:5 anyway) but not ordination to any priesthood. ‘Paul’ does say here that he was ordained an apostle (ordained in the KJV of 1 Timothy 2:7 anyway), but again he says nothing about any priesthood or who it was that ordained him an apostle, if indeed it was anyone but God.
Thus Paul and even ‘Paul’ never mention priesthood or ordination to it.
Furthermore, when Jesus calls the twelve and seventy(two) in the gospel accounts (written after Paul), he is not made to say anything about priesthood or ordination to it. He does ordain the twelve (he ordains them in the KJV of Mark 3:14 anyway), but it is not to any priesthood.
In Acts (also written after Paul), when Matthias is ordained an apostle in place of Judas (ordained in the KJV of Acts 1:22 anyway), it has nothing to do with any priesthood. It has to do with him having been there at Jesus’ baptism, resurrection, and everything in between. That is, Paul would be excluded as a candidate here (but see Acts 14:14). It also has to do with a kind of divination. Matthias is ordained (again in the KJV anyway) as God chooses him by sortition. None of the eleven place their hands on his head to ordain him. Rather, they give a lot to him and lot to another candidate, and the lot falls to Matthias because, it was believed, God made it happen.
Later in Acts, when Peter and John bring the spirit to the baptized Samaritans, something Philip is supposed not to be able to do, there is a power mentioned to be sure. And Peter and John do place their hands on the heads of the Samaritans. But the power they have is not called priesthood. Likewise, when Paul re-baptizes the Ephesian converts of Apollos.Apostle, seventy(two), elder, bishop, deacon are not priesthood offices in the New Testament. Much less do they comprise a priesthood organization in which to become an apostle, seventies first become elders, bishops, and deacons.
The only priesthood organization in the New Testament is the Jewish temple priesthood. Jesus was a Jew but he was not a Jewish priest in the temple; in fact, he seems to have gotten along poorly with most who were.
It is only in the book of Hebrews (which does not even claim to be written by Paul) that Jesus is said to be a priest. There he is said to be a cosmic priest ministering in a heavenly temple/tabernacle. Not exactly literal.
The only occurrences of words for priesthood in the New Testament are in Luke (1:9 Jewish temple priesthood); in Hebrews (7:5, 11-12, 24 one after order of Aaron and another after order of Melchizedek); and in 1 Peter (2:5, 9). It is in the latter that believers in Jesus are referred to as a priesthood. But they offer spiritual, i.e., metaphorical sacrifice. If the sacrifice is non-literal, what about the priesthood?
Believers in Jesus are also referred to as kings and priests to God and Christ (or vice-versa) in the book of Revelation (1:6, 5:10, 20:6), it is true. But it is not clear what that means when God and the Lamb are said to be the temple of the New Jerusalem (Revelation 21:22).
Up shot: there is no precedent in the New Testament for men or women being ordained to priesthood offices of apostle, seventy(two), elder, bishop or deacon. Men and women are referred to as apostles and deacons, the term apostle being used in more than one way by different New Testament authors and probably even the same author (Luke-Acts). Men are also referred to as seventies, elders, and bishops. This has nothing to do with any priesthood or ordination to it. In some New Testament texts, Jesus and believers in him are said to be priests or priestly, but it is only on very broad analogy with the priesthood of the Jewish temple or Israelite tabernacle.