Green’s Dictionary of Slang

to drink.

c.1561		Awdeley Fraternitye of Vacabondes in Viles & Furnivall (1907) 13: Rince Pytcher is he that will drinke out his thrift at the ale or wine, and be oft times dronke.
1820		Jack Randall’s Diary 34: Bid him not turn ’em up for the rincing [...] tell him, my oldone, without any mincing, You mopp’d them out (*drank up the remainder) ere you brought them to him.