Hi, You can only do it as a day trip if you drive, about 2.5 hours each way. With public transport, it's a little over 5 hours each way.
By train, you take the national rail to Catania and then transfer to a regional private train line: Ferrovia Circumetnea or a bus with roughly the same connections at: Bus SAIS Autolinee
Driving in Sicily is not an insane idea and really the only way to do it as a day trip. Use ViaMichelin.com for driving directions and get the extra insurance in case you ding the car.
If you are really nervous and would spend the night in Bronte you could also split it up like this: take the train to Catania, taxi or bus out the airport, pick up a rental car outside the city and drive the 1.5-2 hours to Bronte.
Hope this helps,
Lisa