Irish traditional music is social music, often played in group sessions. Consequently, the genre has standardised on keys and note range in order to allow a collection of different instruments to play together. The basic keys are D, G, and their relative minors. Within these keys, the vast majority of tunes fit a note range of an octave and a sixth, beginning on D and ending on B in the octave above. B is the highest note in the first position on the fiddle, and D is the lowest note playable by a whistle or unkeyed simple system flute.
This range is a problem for single chambered ocarinas, as they only sound about an octave and a fourth, leaving two notes out of range. This means that many tunes, including most of the 'standards', go out of range. There are two ways of addressing this: look for tunes that fit in this range and use a G or D ocarina to play them at the needed pitch, or use a multichamber instead.
There are a decent number of tunes that will fit within an octave and a fourth, many fitting within the D to G range of a single chamber D ocarina. I give a few such tunes below, and more on the page Ocarina friendly Irish music. If you look to other folk traditions, there is a huge volume of music within this range. A lot of English and Breton tunes will fit, and the entire repertoire of the Highland pipes will fit on a G ocarina. Not all of it works musically due to differences in playing characteristics, however.
To play the majority of Irish music on the ocarina, you really need a multichamber. In my opinion, a Pacchioni system double in D is the best suited to this. Having an instrument in D helps as diatonic D instruments are the norm in this tradition. This eliminates the need for most cross-fingering and simplifies ornamentation. The Pacchioni system is also useful, as it tunes the chambers with a note overlap, reducing the need for chamber switching. The chamber break also aligns with the octave break on flute and whistle.
The following Irish tunes will all fit on a single chamber alto D. Note that the tilde refers to a roll, not a classical turn. How to play one is covered on the page Ornamentation: rolls, cranns, and strike cranns. Irish music is fundamentally an aural tradition, and a lot of the details of the music are not indicated anywhere in written music. If you want to play it authentically, you really need to listen to a lot of performances. The basic playing style is legato by default, with tonguing used sparingly to emphasise key notes and separate phrases.