In Luke 5 did Levi become an apostle almost straight away? In Luke 5 we have the conversion of Levi. It appears that he became an apostle almost straight away. Is that what is happening here?
Yes. From that point on Levi became a disciple of Jesus. 

Levi left everything and followed Jesus. The Greek for left everything, katalipon panta, means abandoned all, forsook or deserted. The word for followed is in the imperfect tense, not a one-off action but a continuing state. 

He didn't knock off for the day, he quit his job. He didn't sneak off early, he abandoned his post. That evening there was a party at his home to which Jesus was, of course, invited. Many other tax collectors were present. It was in part a Leaving Do in which many of his former colleagues came to say goodbye. It was also a chance for them to meet Jesus, and to understand for themselves what had prompted his resignation, and the new life he had chosen.

It was not until some time later, perhaps a few weeks, that Jesus chose from amongst his many disciples Twelve in particular. So whether or not Levi became one of the Twelve it did not happen immediately. Long tradition says that Levi and Matthew were one and the same person.  St Augustine of Hippo in his Harmony of the Gospels Book 2 Chapter 26 thought so. In Matthew, Mark and Luke the story of the calling of a tax collector immediately  follows the healing of the paralytic, and all say the calling of the tax collector was followed by a meal at his home, which led to identical complaints from the pharisees. It is widely conjectured that this was one and the same incident.

Luke and Mark refer to the tax collector as Levi, and include the name Matthew amongst the Twelve. If he is one and the same they may be referring to "Levi" as the name he used then, rather as we might refer to an incident involving the Duchess of Cambridge before her marriage as happening to a Miss Middleton. They may also have wished to avoid embarrassing Matthew by mentioning his former profession. Matthew,  in his own gospel, refers to the tax collector as Matthew, modestly having no wish to hide his shady past.   Similarly, whereas Luke refers to a great banquet, Matthew plays down the occasion as a dinner.

Following the Resurrection Matthew became one of the Apostles, as did all the Twelve disciples apart from Judas. The difference between a disciple and an apostle  is that a disciple learns and an apostle teaches. During Christ's earthly ministry they were disciples, afterwards apostles.

So yes, Levi became a disciple immediately.   He probably, though some reasonably doubt it,  became one of the Twelve Disciples  when they were called, a few weeks later. If so,  after the Resurrection he became an apostle.