Hostelling International: For paintings, I enjoyed my visit to "Kunsthistorisches Museum"=lots of paintings from 15th century and later (even the building itself is quite impressive) - Karine (Belgium)
Markus Rach: I recommend the tour through the opera - goes for about 45 minutes - thus doable. It's quite interesting to see the backstage area which is three times as large as the opera guest area...
History Museum · Kärntner Viertel · 73 tips and reviews
Otel.com: Now a museum, this is the place where the composer lived for a couple of years. Located at the Old Town, and just a few minutes away from St. Stephen's Cathedral.