The account of someone from the Kintyre peninsula born in the 1790s ending up farming in Armagh by 1813, sounds a bit unlikely to me. McCullough and its variant spellings is a fairly common name and I suspect different families may have got mixed up here.
The main Scottish migration to Ireland was in the 1600s, when some 200,000 Scots settled there. By the 1700s they were starting to move out of Ireland again and by the early 1800s the number was steadily increasing. That’s not to say that a couple from Scotland couldn’t have moved to Ireland in the early 1800s but it would be bucking the trend. There was certainly plenty of movement between Co Antrim & Kintyre (It’s only 11 miles and boats were back and forth all the time) but in general, apart from obvious commercial trade, it was Irish born people going to Campbeltown to look for work. Not many came the other way because there wasn’t any work in Ireland. That was why folk were leaving. And where would someone from Kintyre get the capital to rent and stock a farm in Armagh? (Most farms were inherited at that time).
I looked at the tithe applotment records for Keady parish for 1825. There was just one McCullagh farm there then. The tenant was John and he lived in Drumdirge townland.
Drumdirge is now known or spelled Drumderg. By the time of Griffiths Valuation in 1864 that farm appears to have passed to William McCullagh. It was plot 3, a 10 acre farm. http://www.askaboutireland.ie/griffith-valuation/index.xml?action=nameSearch
This looks to be that farm in 1901:
The family were Presbyterian (indicating Scottish ancestry) but, as I mentioned previously, most of the Scots settlers in Armagh and the rest of Ulster, arrived in the 1600s.
Neither of the 2 Presbyterian churches in Keady has any baptism or marriage records before 1838, so if that is where the family lived you may struggle to find documentary evidence of their marriage or children’s baptisms.
Possibly DNA testing may be a way of matching with others who have additional information about where the family originate. Family Tree DNA reportedly has more people with Ulster roots than any other company. That obviously increases the chances of finding a match. You might want to try them or, if you have already tested, you can transfer your results to them for no fee.
The North of Ireland Family History Society is running an Ulster DNA project in conjunction with Family Tree DNA and can offer testing kits at a reduced price. http://www.nifhs.org (Go to DNA project on the website). You don’t need to be a member of the NIFHS to participate in the DNA project.