There are as many as 3 transport companies that commute between the Vienna - Schwechat airport and Bratislava:
• RegioJet/Student Agency,
Just take one of them and get off the bus at the "Bratislava, Nivy bus station" stop. It is recommended to book the ticket in advance, because the bus is sometimes completely booked (sold out) in advance. The travel from Vienna - Schwechat to Bratislava Main Bus station takes cca 55 to 60 minutes.
Another possibility to get from the airport to Bratislava is to take a taxi