Cafe Museum is between the Karlsplatz subway station and the State Opera House. Its customers included Ludwig Wittgenstein, the artists Gustav Klimt and Egon Schiele, as well as the Jugendstihl architect Otto Wagner. It's famous for having an interior designed in his typically spartan fashion by Adolf Loos.
This historic cafe was also a stomping ground for such literary heavyweights as Elias Canetti and Robert Musil. Canetti frequently mentions Viennese cafes in his autobiography, esp. in the third volume thereof (The Play of the Eyes). Musil, by contrast, is much more sparing in his cafe references -- in his Diaries he devotes much more space to describing his walking routes through Vienna.
Here's one of Musil's few mentions of a cafe: "We then went on foot to the Cafe Museum in the hope of meeting Morgenstern; I wanted to tell him about Frankfurt. But, to our surprise, we met Frau Krista." (Robert Musil, Diaries: Robert Musil 1899-1942 , ed. Frise & Mirsky, trans. Philip Payne [Basic Books, 1998] p. 365, entry for March 16, 1930)
Canetti's descriptions of Cafe Museum are more informative. Here are three of them:
"Georg Merkel, a painter ..., was a man of about Broch's age. I had seen him at the Cafe Museum, though less frequently than some other painters." (Elias Canetti, The Play of the Eyes, trans. Ralph Manheim [Granta Books, 1999], p. 128)
"At the Cafe Museum, where I went every day after moving back to town, there was a man [later identified as Dr. Sonne] whom I noticed because he was always sitting alone and never spoke to anyone. That in itself was not so unusual, lots of people went to cafes to be alone among many." (Ibid., p. 112)
"I saw [Alban Berg] last at the Cafe Museum a few weeks before his death. It was a short meeting, at night after a concert." (Ibid., p.230)
Update (Sept. 27): There's a nice description of Cafe Museum in Wittgenstein in Vienna (by Allan S. Janik & Hans Veigl [Springer, 1998], pp. 57-8). The authors say that this cafe was originally (1886) at Babenbergerstrasse 5, and then moved to its current Loos-designed location in 1899. Among its patrons, they list (in addition to Musil, Klimt and Schiele) Hermann Broch, Peter Altenberg, Karl Kraus, Joseph Roth, Franz Blei, Roda Roda, etc.
They also include this bit from a letter written by the poet Georg Trakl: "Dear Mr. Ficker! I beg you to loan me 40 crowns, for I am momentarily in a very sad situation.... I would be very happy to be able to meet with you tomorrow, Thursday, at 2 p.m. in Cafe Museum." (Wittgenstein in Vienna, p. 58)