we would like to find a non-touristy place where locals go to, serving Austrian dishes, downtown Vienna
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It depends upon what you mean by 'downtown'; and where Vienna is concerned, the season you plan to visit...
About a 15 minute walk north of the Franz-Joseph train station lies the old village of Grinzing. It' on a hill overlooking Vienna and the Danube. On and off the main street, Cobenzgasse, you'll find an abundance of charming restaurants featuring Viennese cuisine, some of which have "denkmal" status in their own right. You'll be spoiled for choice; find whichever one looks right for your mood and wallet and you'll be all set. Of course if you see a restaurant in Grinzing with a 'denkmal' plaque, there's a solid bet it will be memorable!
During late spring and summer, people eat outdoors in the rear garden.
It goes without saying, when you visit Grinzing, bring a camera.
I can highly recommend the restaurant "Immervoll" (tripadvisor). It is a very hidden place, and easily to walk past if you do not know it is there. It serves excellent food, but is much cheaper than a "gourmet" restaurant. I have eaten there numerous times over years and was never disappointed.
It is also located VERY centrally in Vienna.
A word of "warning": Do not eat an appetizer. The portions are huge and the deserts are exceptionally good. If you eat an appetizer and a main course, you will not make it through the desert. Remember that Austrian kitchen does not use such things as "half fat" cream.
In summer, you can eat outside on the terrace located a bit around the corner on the Franziskanerplatz. Make sure you reserve a table.