Baur started his career with Tirol Innsbruck and played 12 seasons for them, winning four league titles and a domestic cup. In 1997 he went for a short break to Japan and also played a season in the German Bundesliga for Hamburger SV. At 34 years of age, he signed for SV Pasching and after another four seasons there he decided to make another move and switched to LASK Linz.
He made his debut for Austria in a May 1990 friendly match against Holland as a substitute for Kurt Russ and was a non-playing squad member at the 1990 FIFA World Cup. His last international match was an October 2002 European Championship qualifying match, also against Holland. He earned a total of 40 caps, scoring five goals.
Baur was announced as the new head coach of SV Grödig on 7 May 2014, four days before the final match of the end of the 2013–14 season. He begins in the 2014–15 season. He had been the head coach of the reserve team of Red Bull Salzburg from the summer of 2012 to end of November 2013.