Rob,
Armagh does not have a county record office. Most of the likely records for someone born in Armagh are either on-line or kept centrally in PRONI in Belfast.
Jackson’s birth, marriage and the births of his children are all before the start of statutory registration (1864 for births and 1845 for Protestant marriages). So you won’t find birth certificates or a marriage certificate. You might find them in church records but to do that you need to know where they lived and what denomination they were. Otherwise it’s a bit of a needle in haystack.
You could try the pay to view sites such as rootsireland but otherwise it may be a question of searching all the likely church records in PRONI (the public record office) in Belfast.
You say the family may have been Methodists. Methodism took a lot longer to become established in Ireland as a separate denomination than in England. In Ireland there was considerable resistance to separating from the Church of Ireland. It was 1815 before Methodists agreed to conduct their own baptisms. However because of continuing loyalty and other factors, many continued to use the Church of Ireland for sacraments for many years after this date and it was 1871 before all Methodists routinely performed their own baptisms.
For marriages, the earliest ceremonies conducted by a Methodist Minister in Ireland that I am aware of, date from 1835 (Belfast Donegall Square, the first Methodist church in Ireland). However in the mid 1800s there were only a few Methodist Ministers in Ireland (Methodism relied heavily on lay preachers). So the shortage of Ministers contributed to the continuing practice of marrying in the Church of Ireland. In addition, in the early years, many Methodist Meeting Houses were not licensed for marriages so that too contributed to couples marrying in the Church of Ireland.
So to summarise, you are unlikely to find many Methodist baptisms before 1830. Few marriages before the 1840s and only a handful for many years after that. If there are no Methodist records in the location you are interested in, I would search the Church of Ireland instead, as that’s the most likely place to find the relevant event.
Not many Methodist Meeting Houses have graveyards and so they may be buried in public or Church of Ireland graveyards (which are open to all denominations).
I looked in the 1901 census for Co. Armagh (the oldest complete census for the county). There were 155 named Orr. 48 were Presbyterian, 8 RC and the remainder Church of Ireland/Methodist. So statistically, Church of Ireland is the most likely denomination for someone born in 1812, but I wouldn’t necessarily rule out Presbyterian either. (Orr is a Scottish surname and given the location in Co Armagh, Scottish planter origins seem likely, in the 1600s. There aren’t any Orrs in the 1630 Muster Rolls for the county, so I’d assume they arrived after that).
Jackson Orr is quite a rare name. (There are none in Ireland in the 1901 census). I searched the PRONI e-catalogue and found this entry. Obviously it isn’t likely to be your own ancestor but I suppose it might just be a relation.
31st March 1852 PRONI ref: D2005/4/1D
Release of a £148 legacy from Isabella Campbell, Ballyclare, Co. Antrim, executor of the will of Jackson Orr, Madras Artillery, India, to Mathew Lowry, Close Park, Co. Down, and Samuel Lowry, Ballygowan, Co. Down, executors of the will of John Lowry, Close Park, Co. Down.
The document itself is not on-line but you can view it or get a copy in PRONI. The PRONI names index also records this person was a gunner in D Company, Madras Artillery. He was formerly of Ballykeel and his will was probated in 1851. The will itself is lost, destroyed in the 1922 fire in Dublin.
There was also a Jackson Orr born in Gortshalgan in Co Tyrone (adjacent county to Armagh) in October 1888. Sadly he died after 7 days. Here’s that family in the 1901 census. (They were Methodist then):
http://www.census.nationalarchives.ie/pages/1901/Tyrone/Drumaspil/Gortshalgan/1736259/
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1888/02531/1937546.pdf
Researchers in the PRONI area: http://sgni.net
