Ideally after several days in Vienna, we'd like to pick up our rental car on Sunday morning, May 11th, and head to Stift Klosterneuburg for a tour on our way out of town. It looks like S. Klosterneuburg will be open on the day (Pentecost Sunday), but I am a little worried about the rental car locations... we normally use Avis and while the website says they are open Sunday mornings, I am curious about holiday Sundays. We were in Germany on the Monday after Pentecost last year (Whit Monday), and many things were closed (including the EuropCar office). Do most places of business also close in Austria for these 2 days?

This Holiday is known as Pfingsten.
Both Sunday and Monday are Holidays here.

It is likely that some rental places could be closed one or both days.
Check carefully with the agencies.

You can always pick up your car at Schwechat Airport.

We've decided to stay in Vienna until Monday, but it is still a holiday I realize. We were hoping to avoid renting at the airport because of the surcharge. I will try to check from here and see if any of the city center rental locations are open on Whit Monday. Our apartment would be close to the Avis and Hertz offices.

Just a quick note-
The Avis office formerly very convenient on the Ring near to the Opera has moved some months ago to a location a little out of city center.
Hertz office is still on the Ring near to the Imperial Hotel.

Thank you Molker!
Our apartment is on Singerstrasse.
The website is http://www.netland.at/wien/phillips/

I believe it is located within the Ring, or close outside it. I will check into getting our rental car from Hertz since it will be the most convenient to the apartment (assuming the office is open on Pfingsten Monday!)