You haven’t said what denomination the O’Hanlon family was. If Church of Ireland the earliest records for Armagh City start in 1750 (with gaps); if Presbyterian, 1st Armagh has records from 1707 onwards, again with gaps and if RC the records start in 1796.
I searched the on-line records for Armagh on rootsireland but did not find any record of a baptism for Patrick around 1757. Possibly he was baptised in a church which does not have any records for that period. Researching ancestors in Ireland in the 1700s is notoriously difficult and you may struggle to find anything on Patrick and his family.