Unlike Google Maps suggests, the airport bus doesn't end at the Catania railway station (which is a good thing, Italian railway stations are to be avoided whenever possible). The bus only takes 10 to 15 minutes to reach the railway station and then sets out on a circular trajectory through the city center, in one direction only (counter clock-wise). You can find a very helpful map here, rest assured that the route basically covers all the main hotels and sights in the city.
|Alibus Catania airport shuttle map, courtesy of AMT Alibus (I guess)|
Take into account that circling through the city center takes more time, it easily costs some 20 minutes to make the full circle. Don't expect there to be clear indications of Alibus stops, let alone timetables, at the bus stops. There is also a local bus, number 457, which should cost only 1 euro and has a similar route, but I haven't used it so I cannot tell if it's a viable budget option. 4 euro won't break the bank either. By the way, shock absorption it not the strongest feature of Catania buses.
If you fly business class or have a Priority Pass, on leaving Catania you can visit the business lounge. It can be reached by stairs or elevator between gates 5 and 6 in the terminal. Don't expect more than 15 to 20 seats in total, but it is much quieter than below in the main terminal, and you can get essentials such as Coke Zero.