Here’s a link to information on Armagh workhouse:
The good news is that the workhouse admission records from 1843 onwards do still exist and are in PRONI (the public record office) in Belfast. They are not on-line and you either need to go yourself or get a researcher to do it for you. (The admission records for 1839 to 1842 appear to be lost).
Armagh Poor Law Union included the parishes of: Armagh, Clonfeacle, Derrynoose, Eglish, Grange, Keady, Kilclooney, Killyman, Kilmore, Lisnadill, Loughgall, Mullaghbrack, Newry (2 townlands only), Tartaraghan & Tynan.
In Ireland, Church of England is the Church of Ireland (COI). So if looking for the family, you would need to search the COI records for each of those 15 parishes. Again they are not on-line but they are in PRONI. (It’s possible not all the parishes have records. Some were COI records lost in the 1922 fire in Dublin). This link explains what records exist, parish by parish:
If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net