Just done a quick Google search from the Airport to Bratislava Castle, and it looks like about 20 to 25 minutes without traffic. About 40 minutes-ish with heavy traffic.
Thanks so much, that's really helpful. All this information has really added a bit of ooomph to our search for a way to visit Vienna .... sounds ridiculous really, but we don't want to spend the earth on airfares, nor do we really want to have to take two planes each way! Cheers, Chox.
I've visited Bratislava on a day trip. Bus No 61 (which runs three or four times per hour) takes 25 minutes to get to the central railway station, going through mile after mile of drab, Soviet-era housing. (The city centre itself though is beautiful).
However Slovak Lines and Flixbus both operate direct express coach services to Vienna from Bratislava Airport, so there's no need to find your way into the city centre anyway.
PS: If you give me an idea of where you're starting from (so that I can see which UK airports are convenient for you) I'll take a look to see if I can come up with any cheapish transport options that you might not have yet considered.