I believe it was well after Jesus was gone. Paul said he saw a blinding light and heard a voice. That's very convenient for a man who seeks power, and he clearly sought power initially by persecuting Christians. Paul seems a little shady to me. Paul/Saul was a well known and respected Pharisee and a Roman citizen. He was doing pretty well for himself by going after Christians and suppressing the "heresy" of Jesus as the Messiah. Becoming a Christian would have been the LAST thing a man seeking fame and power would have done in first century Jerusalem. He gave up EVERYTHING to follow after Christ as a believer and he suffered greatly, even to death, for doing so. If you are thinking of what a Pope experiences, that was far from anything Paul or Peter, for that matter, was looking for or ever experienced.
As for Paul's meeting of Jesus, I read HERE, that:
There’s no indication from Scripture that Paul and Jesus ever met before the Damascus Road incident. And Acts 9:4-7 doesn’t specify whether the Lord’s encounter with Paul was physical or not. It only says Paul saw a bright light and heard a voice. The men with him heard a loud sound but didn’t see anything. In subsequent re-tellings of the encounter Paul never indicated that He had actually seen Jesus at that time. But Paul did spend 3 years in Arabia where he received the Gospel from the Lord (Galatians 1:11-17). And he made a visit to the Throne of God (2 Cor. 12:1-4) where he saw things he was not allowed to talk about. In 1 Cor. 15:3-8, when Paul recounted all the Lord’s post resurrection appearances, he included himself as one who had seen Him. So, at some point, he apparently did have a physical meeting with the Lord.
Becoming a Christian would have been the LAST thing a man seeking fame and power would have done in first century Jerusalem. Nonsense. There have been many, many people who believe they are chosen, or are the Messiah, or are the second coming, or have some great destiny... it's all ego and mental illness, of course, but it's quite common. And yes, they do seek fame and power. They really believe that supernatural intervention will place them very high, beyond any danger. They generally end up getting killed.