Scots Presbyterians started arriving in Ireland in significant numbers from 1606 onwards, as part of the Plantation of Ulster. Some returned to Scotland, particularly around the time of the introduction of the Black Oath in 1639 and then the native Irish uprising in 1641. Then in 1642 a 10,000 strong Covenanter army under General Munro arrived at Carrickfergus. When that army was disbanded in 1644, many of the Scots chose to remain in Ireland rather than return to Scotland. Further waves of Scots settled in the 1690s due to famine in Scotland. Somewhere between 100,000 and 200,000 Scots arrived in Ireland during the 17th century, most of them in the counties of Ulster (which includes Armagh), representing something like 10 to 20% of the entire population of Scotland.
I looked at the Muster Rolls for Co Armagh c 1630. There are no Pattons in the county at that date suggesting your family arrived afterwards. (There were Pattons elsewhere in Ulster at that time).
There are no records of where and when the migrants arrived. Indeed migrants is also most too strong a term. It’s only 11 miles across from Scotland to Ireland at the closest point and many would have seen it as more like a local move from one county to another, than a migration. Folk went back and forth all the time. For example, it is recorded that in the 1630s, when there were no Presbyterian Ministers in Killinchy, in Co Down, the entire congregation would sail over to Stranraer in Wigtownshire, to attend church and then back again in the afternoon.
For more detailed background reading, “Eagle’s Wings – The Journey of the Ulster Scots & Scots Irish” by Dr David Hume is worth getting.