The use of "a" would mean "She drinks only one glass/bottle of beer a day." However, the Greek is nonspecific. She may drink more than one beer once a day. E.g. She drinks beer at lunchtime...but we don't know how much.
I'm slightly confused as to why "την" is needed in "την ημέρα". Is the definite article always used when describing frequency? Would it also be "μία φορά το λεπτό" and "μία φορά τον χρόνος"?
Exactly, but it would be μια φορά τον χρόνο, because the accusative of (ο) χρόνος is (τον) χρόνο.