Statutory records of births, deaths & marriages started in Ireland on 1.1.1864 (1st April 1845 for Protestant and Non-Conformist marriages). Prior to that you are heavily dependant on church records, where they exist. Most early Irish censuses have been destroyed so no help there. You really need to try and find out a little more about the family from sources outside Ireland. (Otherwise it’s a needle in a haystack).
To use church records effectively, because there’s no centralised index, ideally you need to know the ancestor’s exact denomination, townland and parish in Ireland. Good sources are immigration & naturalisation records which can show town/townland of origin as well as parents and siblings travelling together. Check every possible census. Census returns sometimes include exact place of birth, instead of just “Ireland”. Look out also for other Irish people nearby in the censuses. Immigrants often settled near people they knew from home. Check any military records, old diaries, gravestones, family bibles, wills and obituaries. Marriage certs in some locations (e.g. US States) can include places of birth, parents, etc for bride and groom. Funeral home records can be very informative. US Death certs can show parents names.