Yes, there is a possibility to travel from Bratislava central bus station to Bratislava airport by public transport. But you have to use 2 busses. First you have take a trolleybus number 205 at central bus station, from bus stop "Autobusová stanica". Then at bus stop "Trnavská" change a bus to number 61, which is going straight to the airport.
You can buy a public transport tickets in yellow ticket machines, that are usually situated at the bus stops.