Ring in this idiom is a euphemism for "arse" so the idiom is a variant of to work one's arse off. The earliest usage is from 1992 per Green’s Dictionary of Slang:
work one’s ring off (v.) [var. on work one’s arse off under work v.]
(Aus.) to work very hard.
1992 [Aus] R.G. Barrett White Shoes 6: Porking some bloke’s girl behind his back while the poor mug’s out working his ring off wasn’t actually cricket.
The slang ring meaning "arse" is from 1798 per Green’s Dictionary of Slang:
- as a ‘circular’ part of the body.
(b) the anus, the buttocks; thus ring-snatcher, a sodomite, ring-snatching, sodomy.
1798 [UK] ‘Cropt Comet’ in Hilaria 101: For I’ve been in the milky way, / And Saturn’s filthy ring, sir.
1937–84 [UK] Partridge DSUE (1984) 977/1: late C.19–20.