What did Paul do in Jerusalem while imprisoned by Felix?

Paul was kept under house-arrest but with complete freedom to have visitors:-

Act 24:23 He then ordered the army officer to keep Paul under guard, but not to lock him up or to stop his friends from helping him. [CEV]

This lasted for 2 years:-

Act 24:23 And he [Felix] commanded a centurion to keepPaul, and to let him have liberty, and that he should forbid none of his acquaintance to minister or come unto him.

Act 24:24 And after certain days, when Felix came with his wife Drusilla, which was a Jewess, he sent for Paul, and heard him concerning the faith in Christ.

Act 24:25 And as he reasoned of righteousness, temperance, and judgment to come, Felix trembled, and answered, Go thy way for this time; when I have a convenient season, I will call for thee.

Act 24:26 He hoped also that money should have been given him of Paul, that he might loose him: wherefore he sent for him the oftener, and communed with him.

Act 24:27 But after two years Porcius Festus came into Felix' room: and Felix, willing to shew the Jews a pleasure, left Paul bound.

Historians and scholars have been able to determine when this occurred:-

Paul's captivity in Caesarea, a.d. 58 to 60, during the procuratorship of Felix and Festus, whochanged places in 60 or 61, probably in 60. This important date we can ascertain by combination from several passages in Josephus, and Tacitus

("History of the Christian Church - Volume 1 Apostolic Christianity" by Philip Schaff)

Paul had 2 years during which time he had complete freedom to have visitors and friends from the different churches keep him up to date on what was going on.

Luke would have seen Paul to get biographical details, historical details of the early church, and background for the trial brief he was preparing to be sent to Rome , which we know as the book of Acts which was written by Luke 60-62AD .

He also wrote some of the letters to the churches during this time because he was detained and thus unable to visit them.

The estimated dates of writing are:-

-Ephesians 60-62

-Philippians 61

-Colossians 60-62

-1 Timothy 62-64

-Titus 62-64

-Philemon 60-62

And these are only the letters that we know about : who knows how many others he wrote?

In summary, Paul was freely able to have visitors and friends and write letters as a result of those visits.