I had to spend about two weeks in Austria in order to get a number of visas (which prices, though considerable, are not included here) for Central Asia. It really is a nice city to wander around, with all sorts of cafes and museums to spend time in and during nice weather lots of parks to explore.
Money-saving tips are pretty much the same for all of Europe: eat cheap and stay in hostels. In particular there are doner stands all around Vienna from around 3 Euro, which is way cheaper than a proper restaurant meal.
Also, if you’ll be there fora while, get the longest-term transport card that makes sense. I found out as I was about to leave that they offer a month-long metro/bus/tram card that would have cut my transport costs in half. Oops.
Uncharacteristically, I spent two nights in a proper hotel right as I arrived to Vienna. Sometimes, though, you just need a bit of luxury and comfort for a few days!
Note that if you’ll be using a lot of public transport AND visiting a lot of museums in a short time the Vienna 72hr City Card (€19.90) can be a good deal. Make sure you’re going to use the museum and shopping discounts, though, as otherwise it would probably make more sense just to buy the metro pass independently (€14.50).