There would be little in irish records for this early, even church records are few in Armagh. Protestant or RC would make a difference. Do you know the Parish?
If still in Ireland 1820-1839 you might find them in the
Tithe Applotments just on line with the National Archives.www.genealogy.nationalarchives.ie
with Parish & Townlands
Do you know Rebecca the mother's maiden names?
(No Harrison's, 2 Robert Wilson's Parish Shrule, and
34 Campbell's all in the Parish of Creggan
Two resources for birth records in Northern Ireland:www.emeraldancestors.com
free to view, pay for recordswww.rootsireland.ie
pay - not for profit
Parish Creggan is a Church of Ireland parish and has no birth records listed but Church of Ireland marriages from 1845-1921.