How could the Jerusalem rulers arrest Jews in Damascus?

The Jerusalem rulers had enough clout with the Roman government to extradite Jews from foreign countries such as Damascus and bring them back to Jerusalem for trial, punishment, and often death. Saul went to those chief rulers to arrest synagogue believers and return them for trial.

Les explains that the chief rulers in Jerusalem 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. Saul had evidently pretty much cleansed the homeland, and then he went to these chief rulers. On the road to Damascus, with a heart of absolute hatred for anything contrary to Judaism, he was trying to arrest and bring back to Jerusalem for trial anybody in the synagogue who were believers in Christ. This is what he's talking about when the Lord called him by grace on that road.

How could the Jerusalem authorities arrest Jews in foreign cities like Damascus?

The Jerusalem authorities had sufficient influence with the Roman government to extradite Jews from foreign countries, such as Damascus, and bring them back to Jerusalem for trial and punishment, often resulting in death. This is why Saul could pursue believers in the synagogue at Damascus and arrest them.