Answer: Welcome to Sicily Pia F!
You really wouldn't have six hours by the time you collect your bags, and your flight might be delayed on arrival.
But you could take the Alibus to Catania, store your luggage in Catania (there's no luggage storage at the airport), and then explore the historic centre around the Duomo and via Etnea. The market would be bustling on a Saturday morning.
You might ring Radio Taxi on arrival, and then enjoy the convenience of transport from the airport directly to Atripical for your luggage storage. Then return to the airport via the Alibus or Radio Taxi.
http://www.amt.ct.it/?page_id=2610
http://atripical.it/en/tourist-services-catania/
Vagabonda