You are able to step up the tower of the Mausoleum, but when you have reached the very top of the tower, you will realize it was not really worth all of the effort.
All that you will see besides a few ordinary house-roofs is the golden eagle on top of another tower of the Mausoleum. Spend some time INSIDE the Mausoleum instead, it is full of great works of art and lovely details !
Unique Suggestions: In summer 2005 there were even construction-works going on at the very top of the tower and getting to the windows was almost impossible...
In case that you intend to step up the tower anyway, you better check at the ticket-box, whether the construction-works are finished already !!
Fun Alternatives: Take the lift or the Schlossbergbahn in order to have a much better panorama-view of Graz from the Schlossberg, most probably you will go there anyway in order to see the remains of the fortifications, the old Uhrturm and the great park around of it !!
Graz isnt very touristy anyway, so I really cannot think of any particular outstanding 'tourist traps'.
Picture of Hasnerplatz, just 10 mins walk from the main square.