For the manuscript tradition, see:
https://pinakes.irht.cnrs.fr/notices/oeuvre/15382/
For the edition, see Bibliography.
Discussion
According to the Greek Passion (BHG 1458-1459), the martyrdom and beheading of Paul took place in Rome under the reign of the emperor Nero.
The text itself reveals the date of its composition, specifically the year 458, while it also includes the date of composition of its textual source (i.e. the former version of the same text), which was probably written in 396.
Bibliography
Text:
Migne, J.P., Patrologia Graeca 85, 713-716. (BHG 1458-1459, with further references)