Double negation
Using correct grammar, this means:
We haven't had any time to drink any (that?) beer
It's hard to know what 'that' is referring to out of context, but quite likely 'the' or 'any' would work in place.
I don't think this is actually a double negative. For example, in French to say "I don't know", you say "je ne sais pas." Both the "ne" and the "pas" mean 'no' in a sense, but you have to use them together to convey the sense of 'no'. I think in this case, "ain't... no" is a similar construction. And it just means "no" or "didn't", etc.