I research the Irish Family History Foundation's database. It contains 12,700,000 surviving rcds for abt 65% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800.
Note 1 - Without a KNOWN county, the chance of an ancestor's rcds NOT being in this database is AT LEAST 30%.
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 John Mulloy B 1818 +/- 4 yrs, Co. Longford, with the father's forename unknown, I found 2 rcds: 1817, Granard RC parish, and 1821 Longford RC parish. Father's respective forenames were James and Peter.
For passenger lists, Google ISTG.
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.