Alexander and Bettina's apartment is in a great part of the city, situated just behind the Karlskirche and a 5-minute stroll to the subway at Karlsplatz. This is the second time I've stayed here and both times has been a great experience.
The apartment is on the top floor of a nice building, with a quaint old lift with a seat to get you there. It's extremely quiet- honestly I never heard any noise from neighbours or the street. It's very spacious and well-equipped. Alex and Bettina live next door and were always on-hand for any problems or questions.
I estimate a 10-minute stroll to the Opera house and another 5 minutes to the Hofburg Palace. 10 minutes more to the Stephansdom. The Belvedere Palace is 10 minutes away.
If you like classical music it's easy to get tickets to the Musikverein and Konzerthaus during the season– both are wonderful venues. Vienna Symphony Orchestra are brilliant- Vienna Phil harder to get tickets for. There are many concerts and recitals.
There are many great cafes and restaurants– try Cafe Central, Cafe Griensteidl, right next to Hofburg, Cafe Schwarzenberg and something a little less touristy, Oberlaa.
I would urge you to stay at Alexander's and Bettina's. You won't be disappointed.