Few Parish baptisms records have survived for Protestants
in Co. Armagh. Marriages start 1845. There are only two parish's with records 1796+ and both are Catholic.
The Tithe applotments of 1829-1839 serve as an early
census substitute. No Cuningham in Armagh, but 14 Cunningham's in Armagh. (No Alan so name not handed down)www.genealogy.nationalarchives.ie
The next census substitute is the Griffith Valuation in 1863 in Armagh. Too late for your family but there were still 117 Cunningham's occupying property at that time.
You might try Trinity College in Dublin for their records, they do have a genealogy section Eneclann that might be of help.
Spellings in Ireland are rather iffy and names can be spelled in many ways depending on whether people could write or if someone was writing it for them.
There are only 14 Cunningham's in the british telephone book for Armagh. Good addresses with postal codes.