How could Saul arrest Christians in Damascus when they were outside Jerusalem's jurisdiction?
The chief rulers in Jerusalem had enough influence with the Roman government to extradite Jews from foreign countries such as Damascus. Saul therefore acted under that authority, seeking to arrest synagogue believers in Christ and bring them back to Jerusalem for trial and punishment, often death.
Les explains that Saul had pretty much cleansed the homeland, and then he went to the chief rulers. Those rulers had enough clout with the Roman government that they could actually extradite Jews from foreign countries, such as Damascus, and bring them back to Jerusalem for trial and punishment, and more than often death. So Saul went to Damascus with a heart of hatred for anything contrary to Judaism, intending to arrest and bring back to Jerusalem for trial anybody in the synagogue he could find who were believers in Christ. On that road to Damascus, the Lord called him by grace.