While being held in custody in Caesarea, Paul appealed to have his court case heard by Caesar himself, as was his right as a Roman citizen (Acts 25:12). Then, on his way to Rome, an angel said to Paul
Do not be afraid, Paul; you must stand before Caesar. And behold, God has granted you all those who sail with you. (Acts 27:24)
So we know Paul must have stood before Caesar, although the actual record of it is not found in the Bible.
It was prophesied that he would preach “before the Gentiles and kings” (Acts 9:15), and so his appearance before Caesar was a fulfilment of this prophecy.
Later, when writing to the Philippians from Rome, Paul wrote “All the saints greet you, especially those of Caesar’s household” (Phil 4:22). So it seems he had made friends with those in the household of Caesar while waiting for his case to be heard.
The emperor at the time was the tyrannical Nero, although he was actually relatively sane at this stage of his reign. The empress Poppaea may have influenced him as, according to Josephus, she was “a worshipper of the true God” and probably favourably disposed to Christians.
About 200 years later, Eusebius recorded that “after defending himself successfully, it is currently reported that the Apostle again went forth to proclaim the Gospel, and afterwards came to Rome a second time, and was martyred under Nero.” (Eusebius, Historia Ecclesiastica, 2.22). There are also many other early church writings which say he was released and embarked on new missionary journals (notably to Spain — see Meinardus (1978)).