Rome2Rio lays out a lot of options, none of them attractive from a time standpoint. You really cannot trust Rome2Rio's travel times or frequencies (or fares), though, so it's worth digging down through any of the options you might remotely be interested in to check on the schedules displayed on the websites of the companies operating the transportation.
If you find a rail connection that takes only about 7 hours, I think that would be fabulous--but also unlikely; you've got a distance of about 440 miles, and you'd mostly be on regional trains, not expresses.
I find mixed-mode travel days somehow more stressful than just being on a train (or a bus) all day long. By the time you drive to the airport in Bari or Brindisi and then from your destination airport to Monticchiello, I think it's going to take a lot longer than 7 hours.
Montichiello doesn't seem to have a train station, so aren't you going to need a car while you're there? Perhaps there's an attractive route from Alberobello to Montichiello that also avoids the areas of densest traffic.