I search the IFHF's online database @ ROOTSIRELAND.IE. It has abt 20 M church/civil/census B/M/D records for MOST Irish counties, including Armagh. Records before 1863 become sparse - almost non-existant by 1800. Protestant records are the LEAST likely to have survived.
For John Carney B 1839 (exact), somewhere in Ireland, with a father named James, I found 1 record: Co. Louth. This MAY be for your ancestor. A view of the detailed record (at a small fee) MAY show more useful info.
Since you have so many names/dates, suggest you visit the database and search on your own.