If you do end up going into the Sacher ‘just to say you’ve been there’ etc. They did (do?) have a nice little cosy bar somewhere in the middle of the hotel, off one of the Kaffee und Kuchen rooms.
It’s been 12 years so who knows if it is still there. I was in my early 20s and staying somewhere several magnitudes cheaper, but very pleasantly spent my last couple of hours in Vienna with a cocktail or 3.
Get your Sacher Torte at Cafe Central or Demel.