my brother in law lives in Bratislava and we vist at least 4 times a year. Up until now we have always flown into Vienna with Aer Lingus and got a bus from the airport to Bratislava, it's very easy, and only takes about an hour at most (should be quicker now that the border checkpoint should be gone...thanks to the EU). the €10 shuttle sounds like a great deal, the bus is a little more i think €15.
Bratislava is a great city, with some amazing restraunts and bars. When we go we treat ourselves and stay in the Raddison on the main square....it is the most cental location, and is a seriously top notch 5 star hotel. We usually get it for about €100 per room per night (weekend rate).
In fact we'll be there again in late November, just in time for the Christmas market.
Oh and Vienna is stunning...an amazing city....but some what more expensive than Bratislava.
Enjoy your trip