transportation between Alberobello to Naples International Airport

Bus Alberobello to Monopoli. Train to Bari. Train to Naples. That is what we did.

I think there are direct buses Bari-Alberobello with the Ferrovie del Sud-Est

There is a bus service to Bari named Trulli Link, run by Ferrovie del Sud Est, reserved to people having a train ticket past Bari. I understand it stops in Bari near the railway station, but on the opposite side of the main building (you have to cross all the tracks in the underground passage).

Once you are in Bari, trains to Naples are not too confortable as almost all off them need a change in Caserta. So most connections Alberobello to Naples need about 6 hours plus the time for the taxi in Naples to the airport. Also possible an itinerary through Taranto and Potenza in 7 hours.

Frankly speaking, any chance to move your air connection to Bari airport? You do no realize that Apulia is so apart from other Italian regions until you try to travel without using Bari or Brindisi airport (but Brindisi is unconvenient from Alberobello).

I would also suggest considering renting a car for Puglia. Then you can return the car at whichever airport you want to fly out of.
Puglia is huge. It takes me longer to drive from Vieste, on the Gargano Peninsula, to Santa Maria di Leuca, the southernmost tip of Puglia, then to drive from San Francisco to Los Angeles.

Thank you all.. Irene