This is a good example of a case that requires the indefinite article - she goes not to any old university, but one in Prague (though indefinite because we don't know which one).
In the Czech Republic, can an establishment called "vysoká škola" also be called "univerzita", or is there a difference?
Not every 'vysoká škola' is a 'univerzita.'
Why is it "Zofie je studentkou ..." and not "... studentka ..." ?
I was told instrumental is often use in place of nominative when you mean that the qualification is permanent, or for jobs (she is a teacher : je učitelkou). It may be more bookish with instrumental, also.
I would not explain it this way. You most often use the nominative for some equivalence and the instrumental when A acts / functions as B.
it is an university
Without an I was looking for other solutions
Actually, it is not "an university," it is "a university," as shown above. Although the word starts with a vowel, that vowel sounds like "yu" here and it requires "a" before the word.