Saint Paul was originally named Saul, after the first king of the Jews. But the Jews in Tarsus had assimilated so greatly with the Romans that he was probably given the Roman equivalent of Saul, which is Paul, at birth. He was called Paul in public, and Saul among Jewish gatherings. It was not unusual for Jews who had integrated into foreign cultures to take on a Hebrew name, and an ethnic name. He grew up under two cultures, that of his Hebrew ancestors, and the Greco-Roman customs of his adopted country. He was greatly influenced by the Greek background; Greek was a second language to him; he studied Greek philosophers.
Nothing is certain as to when he came to Jerusalem. His whole family moved there when he was a young man. The year 30 A.D. is as good a barometer as it gets. Scripture scholars claim there was little possibility that he ever saw Jesus during His lifetime. There are others who believe that, while he may never have spoken to the Lord, he may have seen Him before His death. Remember, they really frequented the same circles. They were at different ends of the spectrum, but they were both religious men.
Paul was a Pharisee. He studied under Gamaliel for three or four years. This would have been during the time of Jesus' ministry, 30 to 33 AD. Jesus became very prominent after the arrest and murder of John the Baptist, probably about a year or so into His ministry. The temple area of Jerusalem was always abuzz with rumors about this new Prophet. The personality of Paul was that of a zealot, a nosy body, who had to know everything that was going on, all the time. Also, he was a defender of the Jewish law, which he believed Jesus was breaking. Paul would have agreed with those who considered Jesus a blasphemer and lawbreaker, who should be dealt with accordingly.