There are very few, if any, birth records that exist in Armagh from 1752.Catholic records don't go back that far and Church of Ireland lost a lot of records in the 1922 Dublin fire.I doubt if any of their records went back that far, anyway.
Armagh County Museum does have a number of documents which give details of people living in that time.The museum is in the centre of Armagh city, and is free to visit.I found them most helpful.Most of its records have been filmed by the Church of Latterday Saints and if you go to the 'family search org.'website, you can find details in their library section.
Armagh Ancestry is the heritage organization for Co. Armagh and have computerized a lot of records.They are also based in Armagh city, but do charge for the service they provide.
Both the ACM and Armagh Ancestry have websites, which can be found if you put their names into a search engine like Google.