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There is a bus to the station every twenty minutes or so, takes under half an hour. About a 15 min walk from the station to centre or plenty of buses/trolleybuses although you should be able to change somewhere and cut the journey time down if you get a network map. It was a standard 60 minute ticket from memory for a euro or so.
I passed through there a couple of months ago en route to Vienna and caught the Blaguss service, which stops at what appeared to be a main bus terminal in the centre of Bratislava. It was quite comfortable and better than the normal service buses.
Come out of the terminal and head to the right and you'll see the sign for it.