I research the Irish Family History Foundation's database. It contains 12,000,000 surviving rcds for abt 65% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800.
Note 1 - Without a KNOWN parish, the chance of an ancestor's rcds NOT being in this database is AT LEAST 35%.
Note 2 - Only 1/2 of Co. Cork's rcds are in this database.
Note 3 - The RC church kept control over most parish rcds from Tipperary (north). These are ONLY available thru the Tipperary Family History Research center at a substantial fee (regardless of outcome).
For Cornelius O'Callaghan B 1822 +/- 4 yrs somewhere in Ireland with the father's forename unknown, I found 19 B rcds.
I assume Cornelius died in Ireland. If so, his death rcd is likely at the General Register Office. Contact it at http://www.groireland.iehttp://www.groireland.ie for certificate copies. These rcds often note an exact DOB. Fees apply.
Google the Foundation's name to lead you to this database. Register, opt to be notified as rcds are added and browse. Pls post your search results to my genforum msg.