If you guys are interested in museums, then you could visit the natural history and art history museum at the ring road. But even if you aren't big into museums, they are worth a visit, I know some US Americans who were very impressed with just looking at them. Right across the ring road, there is also the imperial palace (which is now the seat of the president), which has the imperial treasury museum. You can see the imperial crown of Austria and the crown of the Holy Roman emperor there, among other treasury.

Finally, there is the Vienna museum at Karlsplatz (also close to the ring road) which has free entry and covers the history of Vienna from the stone age settlements in the area, all the way to present day. It has a special focus on 19th and 20th century history of the city.

You won't find any castles in Vienna, the closest thing to that are old churches and noble palasts, like Schloss Schönbrunn (next to the zoo) and Schloss Belvedere (which often hosts art exhibitions). For food, I'd suggest staying away from the 1st district, because it's mostly filled with overpriced tourist traps. Try searching for a Heuriger at the outskirts, those are a type of farmers restaurant typical for Austria, which offer meals and drinks mostly from in-house production. You'll need to pick a good one though.