From the discussion forums on Paul and his use of Greek kione and differing stylkes and approaches taken in what were occasional letters.
I suspect Paul might have had a reasonable working knowledge of Latin because he lived under Roman occupation and the language of the dominant power is always the most necessary mode of communication for any oppressed or dominated group in cultural terms. (If we think about people living in Afghanistan or Iraq, the need to understand and be able to communicate in English with the occupying forces would be crucial.) In the same way, many Angolan people are now learning Mandarin in order to be able to communicate with the new superpower China buying tracts of land and involved in trade in East Africa.
More than just being able to communicate in Latin, Paul probably had a good grasp of Roman concepts and thinking, military and religious and civic. This would have bee a survival strategy and a way to help others imprisoned or persecuted by Roman authorities. Paul’s understanding of the law and law courts would have been based on personal experience in dealing with the Sanhedrin and Roman civic authorities.
Since he thought of himself as Jewish, Paul, formerly Saul of Tarsus and someone who had followed Gamaliel, would have had a certain understanding of Hebrew (possibly Aramaic as a spoken language too).
To be multilingual I suspect was far more common that it is in the West now. Most people in Third World countries do speak four or five languages for commerce, for church attendance, for writing or acquired from watching Western TV shows, films or listening to music.