'Northern Ireland' only came into existence in the 1920s, so any earlier reference to 'Ireland' could refer to any part of the Island of Ireland. Any earlier reference to the North of Ireland would likely be a geographic term i.e. the northern part of the island. The province of Ulster contains 9 counties, six of which make up 'Northern Ireland' part of the UK, and the remaining 3, Cavan, Monaghan & Donegal are in the Republic of Ireland - sometimes refered to as "Southern Ireland".
Protestant can refer to a number of different Christian Denominations in Ireland - e.g. Church of Ireland (Anglican), Presbyterian etc, Church of Ireland being the one with largest numbers.
The difficulty with searching for a possible baptism for your James in Co. Monaghan is that not all Church of Ireland and Presbyterian records are available online, and some of the historic Church of Ireland records have not survived. As far as I know Ancestry does not have any significant collection of Protestant records for Ireland.
A quick search for Armstrong baptisms for 1835+-10 in Co. Monaghan on RootsIreland (pay-website), shows quite a number ~190, including a possible match that is probably worth following up - it's the only James baptised with a father Hugh or similar in that timeframe in the records available on RootsIreland :
Currin Church of Ireland parish, Co. Monaghan:
James Armstrong, date of birth 27th August 1837 (baptism date not given)
Minister Robert Woods
Father Hugh Armstrong, occupation labourer, mother Maryanne (CofI baptism dont generally include maiden surname)
(transcript notes that surname is recorded as 'Armstronge' in the register)
Aghnahola is a townland in Currin civil parish, and it covered just over 93 acres, you can view the entry and map on the Placename Database Logainm.ie