Most music in the Irish tradition has a sounding range of an octave and a 6th, D to B an octave higher. Consequently it will not work on a single chambered ocarina. That being said there are a considerable number of Irish tunes in a range we can play. A number of compatible tunes are given on this page.
These tunes are noted in the key they are generally played in sessions. Most require an ocarina in D, a few need a G ocarina. Some of these such as the wedding reel (Macleod’s Farewell) and The Athol Highlanders are actually Scottish. They are commonly played in Irish sessions, however you could not play them in a competition.
A lot of the tunes on this page are recommendations from a thread on 'The Session', a database of folk tunes.
In the next tune play an F sharp in place of the high A.
Would work on a C ocarina, play the low A's up an octave.