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06:22 Apr 30, 2005
English to English translations [Non-PRO] Cinema, Film, TV, Drama / subtitle
Explanation: It's true that 'draw' can mean next in turn for cards, and a challenge in a gunfight to 'draw first', but in a bar this is a play on the latter meaning. To be fast on the draw in pulling a pint of beer.
A 'brewski' is a beer (see first referenced source below).
Of course, the people may not even be cowboys, which would make the joke even clearer. If you provided a little more context this would have been clearer.