I am disabled, so 300m is the absolute maximum, when I move on foot, 50m is of course even better;)
I think you'll find a lot of places that satisfy your requirements: Vienna has many coffee houses and an excellent public transport network. If you pick a random location in the central district, it's more likely than not to be under 300m from a transport stop. With that in mind, it's not really practical to assemble an exhaustive list, so I'll just give an example to demonstrate.
Picking Stephansplatz U-Bahn at random, I see from the Google map that it's about 100m from Café Hawelka in Dorotheergasse, which is a thoroughly traditional coffee house. The same stop is about 50m from a branch of Aida. Aida isn't really a typical coffee house (it's a franchise, for one thing) but it's certainly a solid Viennese institution with excellent coffee and cake, and very popular with locals.
There's Cafe Central, which according to Google Maps is about 100 meters from the Herrengasse U-Bahn station. Someone on Tripadvisor calls it "a must-see for anyone wanting to capture the essence of Vienna."
Go about the same distance in the other direction, toward the Michaelerplatz, and you'll get to Cafe Griensteidl, known in the 19th century as "Cafe Groessenwahn" for all the writers ("megalomaniacs") who would hang out there.
There's a coffee shop called Espresso Schweden directly across from the Schwedenplatz metro stop. AFAIR the stop is above ground, so perhaps the walk would be easier. Not as well-known as Cafe Central, although the ice-cream shop next door has some tradition behind it.