There were small country school scattered all over the countryside in the 1800s. Ordnance Survey Memoirs list school in each area along with year founded, number of pupils, teachers, etc. PRONI have lots of school registers but suspect the ones you need might not go back far enough. School registers list name of pupil, age and date starting school, religion, father's occupation, sometimes (later usually) list exact birthdate and previous school.
Think the first thing you have to do is check what schools were actually in Irvinestown in 1821 (or earlier) then see if mentioned in O.S. Memoirs or if registers at PRONI cover early 1800s.
