Much of Christianity is based on St. Paul's writings. Did Saul of Tarsus ever claim to be a prophet of God anywhere in the new testament?
Paul did not explicitly claim to be a Prophet although there were occasions where he acted as one (cf. Acts 13 & Acts 27), however he did claim to be personally called as an Apostle by the risen Christ Jesus.
The Apostles were specifically set apart by Jesus and given authority to teach and are generally considered by Christians to have either equivalent or greater authority to Prophets in regard to their writings:
Consequently, you are no longer foreigners and strangers, but fellow citizens with God’s people and also members of his household, built on the foundation of the apostles and prophets, with Christ Jesus himself as the chief cornerstone. Ephesians 2:19-20 NIV
And God has placed in the church first of all apostles, second prophets, third teachers, then miracles, then gifts of healing, of helping, of guidance, and of different kinds of tongues. - 1 Corinthians 12:28 NIV
Paul was an apostle, but never claimed to be a prophet of the Old Testament variety. But he did claim to prophecy in the New Testament variety, which is the gift of prophecy. Two different offices/manifestations.
Having then gifts differing according to the grace that is given to us, whether prophecy, let us prophesy according to the proportion of faith;
1 Cor. 13:2
And though I have the gift of prophecy, and understand all mysteries, and all knowledge; and though I have all faith, so that I could remove mountains, and have not charity, I am nothing.
1 Cor. 13:9
For we know in part, and we prophesy in part.
1 Cor. 14:3
But he that prophesieth speaketh unto men to edification, and exhortation, and comfort.
The differences between an OT prophet and a NT prophet are many. For examples, the authentic OT prophet spoke as God directed (thus sayeth the LORD), while the NT believer covets the gift of prophecy. They judged prophetic veracity by results (came true, didn't lead away from the true God). We judge against Scripture and Spirit spiritually. They prophesied of what was to come (primarily Christ). We prophesy to edify, exhort, comfort in Christ.