The most common and most likely answer to this crossword clue is the 6 plus 3 letter word BITTER END
I've seen this crossword clue in The Times
B
I
T
T
E
R
E
N
D
More clues leading to the result BITTER END
Buddy explains!
The straight part
We know that "The critical point at which the beer runs out?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case the critical point at which the beer runs out? = bitter end.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.