I search the IFHF's online database at ROOTSIRELAND.IE. It has abt 20 M church, civil, census, etc records for MOST Irish counties. Records before 1863 are increasingly sparse - almost non-existant before 1800.
For Paul Mongahan B 1841 +/- 5 yrs, anywhere in Ireland (for which surviving records have been transcribed, with the parents' names unknown, I found 0 records.
NOTE1 - The potato famine began in 1845 and ended (to a great extent) in 1850. However, it sparked among the Irish the idea that better lives could be had elsewhere.
NOTE2 - The forename Paul is HIGHLY unusual in Ireland. This is the 1st time I have seen it during years of research. There are 143 Patrick Monaghan in this search range.