I have never come across any official bastardy records in Ireland. They seem to be more common in England, especially prior to the Poor Law legislation.
You could get a copy of the child?s birth certificate to see if the father was named on it. (He will only be named on it if he was present at the registration and acknowledged paternity, but that might have happened). You could also check the church baptism records, as in some cases of illegitimate births the father, or alleged father, is named there.