You always use the genitive after the progressive "ag verbal-noun" structure.
If you do a very crude literal translation as "I am at the drinking of beer", you can see how the genitive makes sense.
There's more information of this issue in the discussion on the exercise "She is opening her books" https://www.duolingo.com/comment/4321993
The 'á' here is really what happens when you put the possessive 'a' (his/her/its/their) immediately after the preposition 'do'.
My understanding (open to correction, certainly) is that we don't have the option to write them separately. The same way that 'do' combines with mé/tú/etc. to give us dom/duit/etc., it automatically combines with possessive 'a' to make 'á'. So "tá beoir á hól agam" really represents "tá beoir do a hól agam," but trying to pronounce that clearly quickly leads one to realize why we combine do+a into a new word.
Because 'beoir' is feminine, we treat the hidden 'a' in 'á' like it means "her [drinking]", just like we would with possessive 'a' meaning "its" where the 'it' is any other feminine noun. That's why we get the h- at the start of 'hól'. If the 'a' were the verbal particle (e.g. from "a bheith", "a ra", "a scríobh", etc.) instead of possessive, we wouldn't get that h- at the start of 'hól'; the initial mutation is a kind of clue to what's going on under the hood.
I keep stressing that it's the possessive 'a' because for a long time, I wanted to think of it as the verbal particle 'a' from "a bheith", for example, but that's a different word.