Ireland at the time was under British rule and Britain didn't adopt the Gregorian calendar until 1752 so it would still officially be on the Julian calender and written records would use this.
Since Catholic countries adopted the Gregorian calender much sooner (it being introduced by pope Gregory) it's possible that Catholic church sources, especially those writing to people on continental Europe might have used Gregorian calender.
Rural areas and 'peasant' farmers (not technically accurate in Ireland) would probably have mostly used saint's days and religious feasts for everyday calender and quarter days for rents and official purposes.
The kingdom of Ireland was still run from Britain, Catholics were excluded from any official office and certainly weren't in charge of making any decisions about calenders. Remember that date/time was vital for navigation and so was controlled by naval observatories with the same sort of military security that GPS has today. Any written history, court proceedings, laws, etc will almost certainly be written using British standards.