Bari to Naples

Take the train. Check tickets for September sometime in June it is too early right now.

Trenitalia
Fast train takes about 4 hours with a change in Caserta
https://www.trenitalia.com/content/tcom/en.html

Last April I used Flixbus to go from Naples to Bari and also used Flixbus to return from Lecce to Naples. I thought it was great. The busses were crowded but affordable. In Bari and Naples the bus stations are next to the train stations. In Lecce, the bus station was in the opposite direction but still walkable from the city center.

Actually the fastest trains are just under 4 hours.
You do have a quick change in Caserta.
Just did that route in October.

I did the flixbus direct from Bari to Naples and it was great. I reserve seat IC, aisle by the door for an unobstructed view. Wonderful.