Czechia has a sophisticated system of bus transportation as well. There are plenty of bus carriers that serve Brno. A comprehensive system for looking up departures of trains and buses all around the Czech Republic can be found here.
There are several bus companies operating buses from Prague to Brno. The yellow RegioJet, Flixbus, Eurolines and other companies operate buses from the bus station Florenc in Prague. The RegioJet buses also leave directly Prague’s Vaclav Havel Airport.
RegioJet and Flixbus will drop you off at the Grand Hotel bus station in Brno (in the city centre), while other companies usually stop at Zvonařka bus station (10 min walking distance from the city centre).
Both RegioJet and Flixbus operate direct buses from Vienna’s Schwechat airport and both stop at the Grand Hotel bus station in Brno. LEO Express leaves from the airport as well but stops at Zvonařka bus station.
Other buses leave from bus stations in Vienna (Wien, Hbf Wiedner Gürtel) — ask for directions at the airport. Check their schedules and buy your bus tickets in advance here.