Bratislava can be conveniently reached from Vienna airport Schwechat by bus, the service providers are companies Slovak lines and Blaguss. Buses run at least one-two times per hour, and they will take you to Bratislava (bus stop "Central bus station" alias "Bratislava AS" alias "Mlynske Nivy" for Slovak lines, about 1 km from downtown; bus stop "Novy most" - standing for a not very new "new bridge" located directly on the historical city border near Danube river) in one hour time.
Round-trip ticket costs a little bit more than 14 EUR (plus an extra charge of 1 EUR per piece of luggage). With Slovak lines you can buy tickets online.
NEW(added note): from certain destinations (Paris-Beauvais, London, Madrid...) Ryanair is operating direct flights to and from Bratislava. Flights can be as cheap as approx. 40 EUR.