The prison he is in is Rome, but was Jews who put him there. It started with his visit to Jerusalem about 57 AD when a mob of Jews started a riot against him. It took a cohort of Roman soldiers to rescue Paul. (Acts 21+) The final upshot of the situation was that for several years the Romans didn't know what to do with him, and Paul, not wanting to be turned over to the mobs, appealed to Caesar, his right as a Roman citizen, and was sent to Rome. This was about 61-62 AD, and Jerusalem was approaching disaster. The Jerusalem church saw their leader James thrown off the highest point in the Temple and clubbed to death at the bottom--they recognized Jesus' warning prophecies and they moved to Pella, on the other side of the Jordan. Paul arrived in Rome probably around 62 AD; the Roman authorities put him under a mild house arrest. He may have been released able to make another missionary journey, possibly to Spain? We do know that he wound up back in a Roman prison, probably about 64 AD. The problem with appealing to Caesar was that the reigning Caesar was Nero.
Nero had become Emperor in 57 AD, but was not a nice person. He believed himself to be a talented singer and harp player. In April of 64, there was a major fire in the city of Rome; raged for days and demolished
There were those in Jerusalem who thought that killing Jesus and Paul would wipe out the threat to their established power; Nero saw this little band of Christians as expendable. Paul and the church were caught between the frying pan of Judea and the fire of Rome; and yet God's people survived, as Paul told Timothy they would.