The earliest census substitute for Ireland is:
The Tithe Applotments 1823-1837 www.genealogy.nationalarchives.ie
5 results for Edward Dempsey, 2 in Kildare Castledermot & Gaulskill Parish's. 1 Kings/Offaly, 2 Ardagh Parish, Longford County.
Griffith Valuation 1847-1864: www.askaboutireland.ie
37 Edward Dempsey's, 10 Haynes mostly from Cork and 1 Kilkenny
Not enough information to do any further searching.www.rootsireland.ie
has baptism and marriages records
for most counties in Ireland, you can find with a birth year and name if the Parish records still are in existance.
Its worth a try.