Vienna don't miss list for 3 days

Hey y’all! We have 3 full days in Vienna next May and I’m trying to build a shortlist before I overplan myself into misery. Must dos on my radar: Klimt at the Belvedere, at least one proper coffee house (Central vs Sperl vs Hawelka??), maybe the Third Man sewer tour because my husband is obsessed with the film.

Is the sewer tour actually cool or a tourist trap? And is Schönbrunn worth half a day or am I better off using that time for the Kunsthistorisches? I’m scared of wasting a whole day on the wrong palace.