This photo is of Doktor-Karl-Lueger-Ring 12, which is by the Schottentor subway station, across the street from the main University of Vienna building (and adjacent to the Palais Ephrussi). There's little of interest in this building (which includes a McDonald's), but I tentatively conjecture that it's where Cafe Schottentor was once located. [NOT! see 2nd update below]
The Schottentor Cafe is one of the cafes preferred by Godel and other members of the Vienna Circle. It is mentioned by Elias Canetti in The Play of the Eyes. He says, "It was amazing how often I ran into Friedl [Benedikt] .... I'd take my seat in the empty No. 38 streetcar, look up, and there she'd be .... She always rode as far as Schottentor .... I went to the Schottentor Cafe. When I entered, she was already there, sitting at a table with friends." (p. 240)
Additionally, in her paper in Endeavour, Maria Rentetzi says that in the 1920's Cafe Schottentor was "a meeting point of Vienna's scientific, political and artistic circles, including women such as the journalist Hilde Spiel." (p. 44, n. 22) Spiel, it turns out, met Canetti at Cafe Schottentor.
It was also a favourite haunt of the poet Theodor Kramer, who is quoted as saying in 1945 that the cafe was opposite the University. Assuming he meant by 'the University' the main university building, the above location is the most likely site of the Schottentor Cafe.
I had initially thought that 'Cafe Schottentor' was an abbreviation of 'Cafe Victoria am Schottentor', the full name of a cafe which used to be at Schottengasse 10 (in the same building as the one pictured below). However, that cafe's name seems instead to have been shortened to 'Cafe Victoria' (which Godel visited with visiting family members, according to John Dawson, Jr.).
Update (Sept. 11): I've learned from Diane Silverthorn that there was another Cafe Victoria in Vienna. It was in the Hotel Victoria on Favoritenstrasse. So, I'm not sure if Professor Dawson's reference is to that cafe or to the Victoria am Schottentor.
Update (Sept. 15): Cafe Schottentor might have been a short block further from the university than I believed. This is according to Professor Karl Sigmund (from the Math Dept. at the University of Vienna), to whom I was referred by Professor John Dawson. Professor Sigmund informs me that there was until recently a Pizza Hut in the Schottentor's old location, but that a new cafe has opened there (at Schottengasse 2). It's called the Café im Schottenstift . The last cafe in that location before the Pizza Hut was Cafe Haag (Hague), which closed in 1997.
Update (Feb. 24, 2009): It looks like my original conjecture wasn't so far off. Here's a pdf copy of a 1936 weekly paper, Die Neue Welt (Feb. 7, 1936), which lists a number of cafes on p. 6 (right column). Cafe Schottentor is listed there as being located at Dr. Karl-Lueger-Ring 10, which would put it just to the right of the above pictured McDonald's (the address for which is Dr. Karl-Lueger-Ring 12). This location is now the site of a cafe called Coffee Day. The building in which it's located looks like a relatively new annex joined to the older structure that houses the McDonald's. I wonder if the part of the structure that housed Cafe Schottentor was wrecked during the war, with the result that the cafe re-located to Schottengasse 2.