Did St Paul and Peter die on the same day?

We do not know when or how Paul and Peter died, but it is most unlikely that they died on the same day.

A tradition grew up around the supposed martyrdom of Peter after the mid-2nd century pope Anicetus (156-166) said that Peter was beheaded by Nero in Rome. The reason for this claim was that he had became locked in a conflict with the venerable Polycarp of Smyrna. Polycarp had tried to win the argument by insisting that he spoke with the authority of the apostle John. In response, Anicetus told of Peter's beheading and said that he spoke with the authority of Peter. Then in the third century, the Church Father Origen changed the story somewhat, saying that Peter, condemned to crucifixion, felt himself unworthy to be crucified the same way as his Lord, and so chose crucifixion upside down.

In fact, we do not know whether either Peter or Paul was martyred, so we do not know how or when they died.