It's from a novel of 1938 by E.C. Bentley's Genuine Tarbard. Now the expression of 'cat's back' seems not to be in use any more. However, what's the meaning of this 'cat's back'?
It's a saying that means something is very high or very tall. It's just one of many such colorful phrases. It references the fact that when cats get scared they puff up and arch their backs.
Similar to phrases like "colder than a witch's tit" or "dumber than a bag of hammers" or "deeper than the holler."
In this case, it means an expense. Example:
“To go up like a cat’s back.” (To rise quickly as the cat’s back docs when the animal is angry.) “The price of corn went up like a cat’s back.” -A word-list from "Bill Arp" [pseud.] and "Rufus Sanders" [pseud.]. by Margaret Gillis Figh. ...A word-list from southern Kentucky, by A. P. Dalton. The secretary's report. (1950)