Fischer, 65, is a Social Democrat who was elected to the mainly ceremonial post of head of state in April. He is a lawyer by training who is married with two children.
Fischer's predecessor Klestil, 71, died Tuesday of multiple organ failure just two days before he was due to step down.
His funeral is due to take place Sunday.
U.S. President George W. Bush will be represented at Klestil's funeral by the state of California's governor, Arnold Schwarzenegger, who was born in Austria and was a friend of Klestil's.
Klestil was widely praised for helping Austria come to terms with its Nazi past. Jewish leaders were among many around the world who sent their condolences upon hearing of Klestil's death.
Celebrity Families of 2014 [PHOTOS]