Hi
Dublin was divided into 4 Poor Law Unions, North Dublin, South Dublin, these would have been mostly city areas, and Balrothery which would now be North County Dublin, and Rathdown which would be South County Dublin.
Catholic births would not have been registered with the civil authorities until about 1865, COI about 1845, but church records can go back a long time.
Finding someone born before 1865 will be difficult and it would help if you could narrow down the search a bit. Do you know any place names or church names that might be spoken off in your family?
Regards