‘Tig le’ in this sense is used impersonally. The verb paart is in the third person, subject unexpressed. The prepositional pronoun ‘le’ will vary, of course, according to person: ‘Tig liom, leat, leis’ etc. – I, you, he, etc. can. Future: Tiocfaidh liom, léi, linn’ etc. Conditional: ‘Thiocfadh liom, leat, libh, leo’ etc. Past: ‘Tháinig Iliom, leat, libh’ etc.
It’s not exclusive to Ulster, though it’s certainly used more frequently there. And in Ulster ‘tig’ tends to be lenited in the present affirmative: ‘thig liom fanacht’ rather than ‘tig liom fanacht’.