I had to google when exactly that day was (June 7th, right?).
Stores will definitely be closed, but I am assuming
Vegetasia (all you can eat taiwanese buffet), http://tupalo.com/de/wien/vegetasia-ungargasse
and it's cousin restaurant
Xu's Cooking: http://tupalo.com/de/wien/xus-cooking
will be open. They are always open!
is open on holidays from 10 am to 6 pm. It's on the pricey side (you pay per weight) but vegan options are clearly marked on the buffet.
will be open, here are prices and opening hours: http://www.mqw.at/en/visitor+service/ho ... titutions/
(I like the MUMOK and Kunsthalle best but if you are into Schiele and Klimt, go visit the Leopoldmuseum).
Albertina will be open too: http://www.albertina.at/
I am assuming there will be some sort of big Catholic spectacle going down by St. Stephen's (Stephansdom) but Austria is a Catholic country, so most churches will have a procession.
Let me know if you have any other questions.