After his school years at the Stiftsgymnasium Wilhering, Klambauer studied flute and piano at the Brucknerkonservatorium Linz and at the University of Music and Performing Arts Vienna with Wolfgang Schulz. During his studies, he was a solo flutist in the Gustav Mahler Youth Orchestra, Camerata Salzburg under Sándor Végh, and in the Youth Union of the European Union under Claudio Abbado. In 1992, he became a member of the Bühnenorchester of the Austrian Bundestheater. From 1993 to 2013 he was a solo flutist in the Radio-Symphonieorchester Wien. Since 2014, Klambauer has been the solo flutist of the Wiener Symphoniker.
Klambauer regularly performs with ensembles Kontrapunkt, the series, Halil and the ORF wind quintet.
From 2004 to 2008 Klambauer taught at the Konservatorium Wien and at the University of Music and Performing Arts Vienna. Since 2008 he has been professor for flute at the University of Music and Performing Arts Graz.