Derivation of "anus" from "annulus"?

You have the derivation slightly backwards. The Latin anus meant ring and thus also the anatomical anus.

Meanwhile anulus was a Latin diminutive of anus, so a small ring

Both anus and annulus comes from Latin, where anus meant "a ring," and annulus meant "a little ring." The Latin annulus is derived from the Latin word anus.

The origin of anus is 16th century, and the origin of annulus is Middle English.

Online Etymology Dictionary has it "from PIE base *ano- 'ring.'" Interesting.

Annulus is also related to annual, or the yearly cycle. Related to Anno Domini - Year of Our Lord. The double 'n' seems to be correct historically although they have a common origin.

Not necessarily surprising that anus may not have been named until the 16th C. The hip bone ("innominate") is the "unnamed" bone. The pudendal nerve (the one that goes numb if you've been riding a bike for too long) means "that which is not to be discussed" and is also related to the slang "pud".