Would she have been someone you'd want to go out to coffee with, or was she one of those people who would spend the whole time obliquely critiquing your every word?
Puritanism was a religious movement in seventeenth-century England, so a "puritan" was a variety of English Christian, like a Pentecostal would be today. Puritanism came about as a result of the rather unsatisfactory way in which the English Reformation unfolded.
King Henry VIII wanted a divorce, and the Catholic Church would not grant it, so he divorced his nation from the Catholic Church instead. And so England became a Protestant nation, following in the steps of other nations in northern Europe. Now: the Reformation in those other European nations -- in places like Germany and Switzerland -- was at its heart a deep evaluation what Christianity was really all about, and how it should be properly practiced. It resulted in churches that operated much differently than the Catholic church. But Henry VIII wan't that into church reform. He just wanted to marry Anne Boleyn, and have some sons -- and maybe get ahold of some of that prime church land, and make some reform-minded (but powerful) clergy happy. The church he founded -- the Anglican Church -- was very Catholic in its services and structure. And some folk in England thought that was a problem.
It was too Catholic. It was full of practices that had no basis in the Bible. The cry of the Reformation was "sola scriptura" -- Scripture alone. Many in England called for the "purification" of the English church of all Catholic influences, of all practices and doctrine that do not have their source in the Bible. In time, those who set themselves to that task were given the name "puritan." It was a derogatory name even then, given by their critics. Puritans called themselves "nonconformists."
So this was not a geezer's movement, not a movement of geriatric church-goers railing against modern life. This was a young man's movement, and a young woman's movement. It was full of zeal and passion. It was an educated person's movement. Certain universities were hotbeds of puritanism. You would send your son to college he might come home a white-hot puritan, and you'd be about as thrilled as more modern parents were when their college kids came home anarchists, or flower children, or Packers' fans.
None of this answers our question about Anne's drinking habits, but we'll get there. One last note, though: I intentionally do not capitalize the word, "puritan" for a reason: it was a movement, not a denomination or a statement of doctrine. Within its borders was a general agreement on certain important issues, but there was disagreement and variety in the ranks as well. Hopefully you're getting that from my clever use of the lower-case.