"She studies at a university."
Translation:Uczy się na uniwersytecie.
Actually to hopefully clear up this point as described my my Polish teacher. "W" is not necessarily for smaller places although you're on the right track. "W" is used to refer to a specific building or place. "Na" is a more general term. For example "Na uniwersytecie" (being at a university is a very broad term) but to say you're in a class, specifically"w klasie". Other examples to attempt to carry the point across : "Na koncercie" but "w klubie", "na zakupy" but "w sklepie", "na spacer" but "w parku". There are some exceptions e.g "na ulicy" for the reason that you are literally on the street and not inside it! Hope this helps!
But also "na poczcie" (at the post office) and "na stacji"/"na dworcu" (at the station, despite the fact that it may be closed space, at least "dworzec" because "stacja" often isn't... "na lotnisku" (at the airport) although that's also a building... the general rule you stated is right, but the number of exceptions isn't that small.