Green’s Dictionary of Slang

(Aus.) to get drinks on credit.

1900–10		Stephens & O’Brien Materials for a Dict. of Aus. Sl. [unpub. ms.] 41: CHAINING UP A PUP: Aust. to run up a score in a public house or store. A pup chained up means an account standing which the debtor is unable or unwilling to pay. If you leave a pup long enough he will become some day a dog mad enough to bite you.
1941		Baker Popular Dict. Aus. Sl.