Personally I wouldn't rely on a bus to Catania airport for your flight
since you have no wiggle room.
I'm not sure I understand this comment. There are multiple buses scheduled to get in within 2 hours of the flight. It's like any other public transport - yes, you do have to time the bus, but you don't only get a single chance. Plus, a taxi is always a last resort option.
I plugged in a random dummy date of August 13th and got the following:
Departing Giardini-Naxos at 6:25, 7:55, 8:25.....
Arriving Aeropuerto Catania at 7:45, 9:10, and 9:40...., respectively
One-way fare is 8,20 EUR
I would check the schedule on your dates because it may vary. If budget is an issue, I would not write off the bus. I depended on it for many of my trips in Sicily and it was predictable once I had the schedule. The only day buses and trains were severely curtailed was on Sundays, so definitely keep that in mind.