When did Paul first see Jesus?

It's common for scholars to deny that Paul knew the historical Jesus. But is that correct?

Paul witnessed the martyrdom of Stephen:

12 And they stirred up the people and the elders and the scribes, and they came upon him and seized him and brought him before the council (Acts 6:12). 

58 Then they cast him out of the city and stoned him. And the witnesses laid down their garments at the feet of a young man named Saul (7:58). 

8 And Saul approved of his execution. And there arose on that day a great persecution against the church in Jerusalem, and they were all scattered throughout the regions of Judea and Samaria, except the apostles. 2 Devout men buried Stephen and made great lamentation over him. 3 But Saul was ravaging the church, and entering house after house, he dragged off men and women and committed them to prison (Acts 8:1-3).

Stephen was martyred about a year after Christ was crucified. Cf. Eckhard Schnabel, Acts, 43. As another scholar notes:

The fact that witnesses laid their clothes at Saul's feet suggest that he was already the acknowledged leader in the opposition to the earthly church (cf. 8:1,3). David Peterson, The Acts of the Apostles, 268.  

In 4:35,37; 5:2, lying something at someone's feet implies a recognition of that person's authority. Ibid., 268n89.

So Paul was in Jerusalem a year after the Crucifixion. And he was a seminal figure in the persecution of the early church. 

What was Paul doing in Jerusalem at that time? He was a rabbinical student. 

I am a Jew, born in Tarsus in Cilicia, but brought up in this city, educated at the feet of Gamaliel according to the strict manner of the law of our fathers, being zealous for God as all of you are this day (Acts 22:3).

This statement indicates that although Tarsus was his hometown, he was already living in Jerusalem well before he began his rabbinical studies. As one scholar notes:

Paul mentions his place of origin only briefly and moves on to his early life in Jerusalem. There are three participles in the Greek, highlighting successive stages in Paul's experience: he was "born" in tarsus, "brought up" in Jerusalem, and "educated" at the feet of Gamaliel. So Paul's theological roots were essentially Palestinian rather than Diaspora Judaism…This phase of Paul's education probably began some time after he turned thirteen, when he was instructed by Gamaliel "according to the strictness of our ancestral law," ibid. 597.

One possibility is that Paul's father moved the family to Jerusalem to give his son proper socialization in traditional Judaism. This preceded his formal rabbinical education. Or perhaps there was an aunt and uncle in Jerusalem who hosted Paul. That's reinforced by another statement: 

16 Now the son of Paul's sister heard of their ambush, so he went and entered the barracks and told Paul (Acts 23:16).

As one scholar notes, 

Paul and his sister were apparently brought up in Jerusalem, and his sitter remained there to marry and have children. Ibid. 621. 

The upshot is that Paul was in town at the time Jesus made trips to Jerusalem during his public ministry. That's reinforced by the fact that Jesus went to Jerusalem during major festivals. 

This would also help to explain why Paul was on the groundfloor of opposition to the Christian movement. He was a protege of Gamaliel, who was, in turn, a member of the Sanhedrin. Gamaliel was directly involved in formulating a policy to counter the nascent Christian movement in Jerusalem. 

26 Then the captain with the officers went and brought them, but not by force, for they were afraid of being stoned by the people…34 But a Pharisee in the council named Gamaliel, a teacher of the law held in honor by all the people, stood up and gave orders to put the men outside for a little while (Acts 5:25,34).

This gave Paul access to official proceedings via his mentor. That explains his presence at the martyrdom of Stephen. By riding on the coattails of his mentor, Paul had entree to the high priestly entourage. But Paul took a harder line than Gamaliel. 

Considering all these connections, it seems quite likely that Paul saw Jesus preach. He certainly had occasion to see Jesus preach. And Jesus drew big crowds. Surely Paul would be curious.

That would explain why Paul took such an early and avid–albeit hostile–interest in the Christian movement. It didn't happen overnight, right after the apostles began preaching the Resurrection. 

That, of course, wouldn't make him a believer. For one thing, that doesn't mean he saw Jesus perform miracles.

And even if he did, the Jewish leaders had an apologetic to explain away the miracles of Jesus: they called him a sorcerer. We find that allegation in the Gospels and Talmud alike. Assuming he saw Jesus preach, he probably viewed Jesus as a false prophet or Messianic pretender. 

But the radiant postmortem appearance of Jesus to Paul on the Damascus road was too much for Paul to deny. Rumors about the Resurrection turned out to be true after all!