fair suck of the sauce bottle

The story

An elaboration of fair go (used in protest to implore or demand that someone act with more fairness or reason). Sauce bottle appears to refer to a bottle of cheap beer or another alcoholic beverage; see sauce (“(slang) alcohol, booze”).

First known use, by meaning

• from 1970s A fair chance, a reasonable opportunity; a fair go.
• from 1970s Used to protest against unreasonableness, such as somebody taking more than their fair share.