I loved Vienna & can't wait to get back, but was there in the summer.

Lots of museums along the Ring, the Vienna Opera House, the Palaces for sure, the occasional cathedral. I popped in a couple of Mozart's apartments (I think he moved every few months), and was never disappointed.

Lots of other musicians to poke around about there, from Schubert to Mahler. This time of year, you should be able to take in some real concerts, though I've heard the tourist ones can be good at times. (If a guy in a white Mozart wig is selling tickets, that's a big clue it's a tourist concert). Vienna was ground zero for a lot of classical music.

I missed Freud's place, but understand that is pretty fascinating.

I enjoyed some of the parks there, including around the Danube, but I'm guessing that won't be a good option for you. Plus, the Danube is quite brown, rather than blue.

You have some time, so I'd suggest reading up on famous people who lived & worked there that interest you, and plan to explore their haunts.

Public transit there, busses & subways, work very well, as with most european cities.

Have a great trip!

JP