The gorgeous Michaelertor can be found in the Hofburg -- as you pass through on your way to the Ringstrasse...
The Hofburg was the winter residence of the Emperor. It is situated in the inner city (First District).
Stroll over the place and you will noctice that i's very nice and beautiful in here. The picture is in the Heldenplatz.
Visit the kaiser appratments in Hoffburg and the gardens belonging to it.
Interesting museum and nice collection of gold and procelain.
An imposing set of buildings containing a number of exhibits, including the crown jewels. I haven't been in years... but that's no excuse for you!
The famous Vienna Hofburg, best visited by classical Concerts (W.A. Mozart, J. Strauss, etc.)
unfortunately without H.v. Karajan...