If you want to sound natural, B is correct. I can't really give a good reason why, though, just that... nobody really says "much" in this sort of context. A lot of means there's a large quantity. "I drink much coffee" isn't WRONG, it's just... usually it'd be more like "too much," as in, you drink an excessive amount.
I think maybe much is used for less easily quantifiable amounts of things. Such as, "we feel much gratitude," or, "there's much joy in the air."