The Bible does not give any statement about when Paul was born or how old he was when he died. We have a little information that may allow us to make some educated guesses, but even those are still guesses. A quick internet search will yield answers that range from a few years before Jesus was born to a few years after.
Paul (Saul) comes on the scene in the New Testament in Acts 7:58. When the crowd was stoning Stephen, Saul watched their cloaks. In Acts 7:58, Saul is described as “a young man.” This does not tell us very much other than that he was obviously not elderly at the time. Perhaps it makes sense to think of him as a young, up-and-coming star among the Pharisees, not yet old enough to be on any official council. He watched the cloaks, but he did not participate in the actual stoning. (In Acts 26:10, Paul says that he cast his vote against believers when they were put to death. The word does not necessarily imply that he had any official vote but rather that he approved of their deaths. He arrested believers on authority of the priests, and when they came before the official council, he approved of or even perhaps advocated for the death sentence.)
The best date that can be affixed in the life of Paul is the event in Acts 18 where the Jews brought charges against him before Gallio, the proconsul of Achaia. The dates for Gallio’s rule are somewhat tenuous but are probably somewhere between 50 and 54. So by AD 54 Paul would have had completed two missionary journeys, spent time ministering the gospel in Judea and Antioch, and spent three years in the Arabian Desert (Galatians 1:11–20). Prior to that, he would have spent time persecuting the church after spending some years studying under Gamaliel. Gamaliel was active for the first half of the first century, so that does not give us any more information. There seems to have been plenty of time for Paul to do all of this if he was born any time in the first ten years of the first century.
If Paul was born about ten years after Jesus, he would have been 20–23 years old during Jesus’ public ministry. He would not have been old enough at the time to exercise leadership in the Jewish community. Assuming the stoning of Stephen was a few years after Jesus’ earthly ministry, Paul would probably have been nearing the age of 30, when leadership positions would begin opening up for him. It might even be that Paul’s campaign of persecution was designed, in part, to make a name for himself—and he could assume a leadership role when one became available. We have no clear biblical support for this, but there is an indication in Romans 7:7–8 that Paul struggled with covetousness prior to his conversion. If Saul coveted power, his persecution of the church may have been a means to attain it.
After Stephen’s death, Saul launched a campaign against the church. He was not yet the one in charge, but he was given permission from the Jewish authorities. He waged this campaign until he was stopped cold on the road to Damascus, confronted by the risen Christ (Acts 9). If this happened when he was not yet 30 years old, perhaps this would help explain his three years in the desert. Once he had finished his time with the Lord, he had received further training in the gospel and was old enough to be a respected member of Jewish society.
The idea that Paul was born about ten years after Jesus is simply one theory that seems to account for a lot of the available information. No doubt other theories could be presented, supported, and even debated. It would be an interesting tidbit to know when Paul was born. However, even the most important birth in history (that of Jesus) cannot be pinpointed to a specific year, much less a specific date (December 25). The kind of precise record keeping that we expect today is a modern preoccupation and was simply not a priority at the time the New Testament was written. We get into trouble when we take a dogmatic stand on questions that the Bible does not answer. Speculation is not necessarily helpful unless such speculation helps get us into Scripture to find out what it really says.
Attempting to ascertain when Paul was born can be useful if it motivates one to thoroughly study all that the New Testament has to say about Paul. This would be of great benefit even though the question itself remains unanswered.