What made him 'do it', or how was Paul uniquely suited for his mission…
What was not happening with the Way or Christian movement when Paul 'came along'? It was not being spread outside of Palestinian Jewish circles to any appreciable degree, or at least to any degree that has survived to present itself to historical investigation. Hellenistic Jews were an oppressed minority within the Jerusalem Christian community (Acts 7,8), so you can imagine that gentiles were 'off limits'. Further, it would seem that after an initial period of evangelical speech-making, missionary efforts had cooled and the community had turned inward and developed an apocalyptic mentality.
The original Apostles and disciples were Palestinian Jews who were once thought to have been lower class people (but evidence exists that this is not the case). Their view, however, was parochial. Do you recall seeing any of the 'gag' maps of New York City that show the city occupying perhaps three fourths of the continental US, and place every major landmark in close proximity and small scale to the city? This was the view of the Jews of Palestine with respect to Jerusalem. To them, Jerusalem was the only really important place in the world, and you did not concern yourself with venturing far beyond it. Further, though they probably spoke some of the colloquial Greek (Koine), they were not well versed in it, and certainly were not suited to communicate to the Hellenistic gentile world, when they could not really embrace the Hellensitic Jewish minority within their own community. Jesus' "Great Commission" had said to "go into all the world", but however gifted the original Apostles and disciples had been, they were apparently not suited or at least not ready to carry this out.
Paul (nee Saul of Tarsus) had grown up in Tarsus, a very important city in Asia Minor. He was Jewish, but lived in the Hellenistic world. He knew the scriptures and the traditions, but was also comfortable with and in the Greek world of thought and culture. Paul's writings are chock full of Greek thought and culture, from the philosophies of Stoicism, Platonism and the Epicurean to illustrations drawn from Greek culture and sport (which were considered 'unclean' by the more conservative Palestinian Jews). Paul was also idealistic, and very zealous. He was initially opposed to the "Way", and brought considerable energy to its persecution. He was "willing to travel" to accomplish his mission. All of these qualifications made him an ideal candidate to become the missionary to the gentiles and the architect of much of what we know today as the church. He supplied all of that which the original Apostles and disciples lacked, and was virtually forced to operate entirely outside of the Palestinian Jewish community, both by his previous career persecuting these people and his distinctly Hellenistic culture and world view.
Paul was the ideal 'candidate' for the role he ultimately filled so well.