Q. When and how did Paul become an apostle? Was Paul an apostle or disciple? Or both?
I think we can say, on the basis of Paul’s own testimony during his trials as they are recorded in the book of Acts, that Paul became an apostle when Jesus appeared to him on the road to Damascus and sent him to proclaim the good news to the Gentiles. The word “apostle” literally means “one who is sent,” so the moment of Paul’s “sending” is the moment when he became an apostle.
When Paul was on trial before King Agrippa, he told him, “I was going to Damascus with the authority and commission of the chief priests. About noon, King Agrippa, as I was on the road, I saw a light from heaven, brighter than the sun, blazing around me and my companions. We all fell to the ground, and I heard a voice saying to me in Aramaic, ‘Saul, Saul, why do you persecute me? It is hard for you to kick against the goads.’ Then I asked, ‘Who are you, Lord?’ ‘I am Jesus, whom you are persecuting,’ the Lord replied. ‘Now get up and stand on your feet. I have appeared to you to appoint you as a servant and as a witness of what you have seen and will see of me. I will rescue you from your own people and from the Gentiles. I am sending you to them to open their eyes and turn them from darkness to light, and from the power of Satan to God, so that they may receive forgiveness of sins and a place among those who are sanctified by faith in me.’ So then, King Agrippa, I was not disobedient to the vision from heaven. First to those in Damascus, then to those in Jerusalem and in all Judea, and then to the Gentiles, I preached that they should repent and turn to God and demonstrate their repentance by their deeds.”
So that was when Paul became an apostle. As for whether he was also a disciple, yes, he was. The word “disciple” literally means a “learner.” In the context of the New Testament, it refers to someone who is learning from Jesus how to live in the way that God wants. In other words, it simply means a follower of Jesus. We sometimes speak of his “twelve disciples,” meaning the twelve men he chose to teach in a special way, and after his resurrection, when he sent them out to proclaim the good news, they became the “twelve apostles.” (One of them, Judas Iscariot, betrayed Jesus, and so he was replaced by another man, Matthias.) But that is a specialized use of the term “disciples.” Generally, it applies to any follower of Jesus, and so it also applies to Paul.