The only direct service to Catania Airport from Messina is the SAIS bus, departing from Messina’s Central Train Station (below the Google Maps location).
You can check the schedule here: SAIS autolinee. The ticket office is in Via del Vespro, just a stone’s throw from the Train Station entrance.
Alternatively, you can get a train to Catania’s Central Train Station which will cost you between 6 and 12 euros; the train takes about 1 hour and 40 minutes. You can then hop on an AMT Alibus to the airport (buses run from 5:00AM to midnight every 25 minutes) or get a taxi if you’re short of time.