Irish Gaelic fáinne was an alternation of áinne, from Old Irish ánne. The Proto-Indo-European root is *āno- "ring", which became Latin ānus "ring, anus".
a ring, Irish fáinne, áinne, Old Irish ánne, *ânniâ; Latin ânus, English annular.
The Early Irish Glossaries Project in the entry for áinne "ring; circuit; anus" has "Cf. fáinne", but no word on how they are exactly connected.
Where did the /f/ come from? I know Celtic languages have a habit of losing initial /f/, but this one was added.