I searched the Irish Family History Foundation's online database at ROOTSIRELAND.IE. It has abt 20M B/M/D records for most Irish counties, including Armagh. Records before 1863 are increasingly sparse: almost non-existant before 1800.
For Ann(..) Moan B 1856 +/- 5 yrs, somewhere in Ireland, with the parents' names unknown, I found 111 records. When I use a father's forename of Ste(...) and mother's of Mar(..), I find 0 records.
Records for Armagh aren't that great. Occasionally more are added (mostly protestant). Suggest you visit the database, register (free), browse (free), and opt (also free) to be notified as records are added.