Ar mhaith leat cupán tae?
The response would be Ba mhaith liom ("Yes""/I would like") or Níor mhaith liom ("No"/"I would not like"). (You can drop the liom in the response, if you want).
I think you're probably more likely to encounter Ar mhaith leat cupán tae? than An bhfuil cupán tae uait?
Why is there no "de" needed in this sentence for "cup of tea"? If it isn't needed, why not, and how is there a distinction between "cup of tea" (with tea in the cup) and "tea cup" (a vessel generally used for tea or the type of cup, not necessarily containing any tea at the moment)?
The "of" in "cup of tea" is an example of the genitive case, which in Irish causes the 2nd noun in the phrase to be in the Tuiseal Ginideach or genitive. As it happens, most of the words that you might get in a cup have genitive forms that are the same as the nominative (tae, uisce, bainne, caife, siúcra) but here are some examples using more obvious examples of the genitive:
cupán plúir - "a cup of flour"
cupán rise - "a cup of rice"
cupán airgid - "a cup of silver". Note that this isn't a regular cup that is filled with silver, but a cup made of silver ("a silver cup"), though, just as in English "a cup of silver" could mean a regular cup full of silver, but that wouldn't be the normal meaning.