I suspect Murray is a common name so narrowing down the where in Ireland will take some work.
I'd start with marriage records and death records here in the US. What you want/need is the parent's names (i.e. Father & mother's maiden name).
You'll find that a lot of the Irish records got destroyed during the 1920's (a long story). The Catholic church was the majority (population) but persona non gratis for government records (civil registry). As a result the church records are spotty, mostly 1820-1888 (though there are lots of exceptions).
The Catholic parish records are on-line via the National Library (registers.nli.ie) but you need a Parish, they aren't indexed.
FamilySearch and Ancestry have some Irish records. There are numerous Irish sites but they usually require a subscription.
You'll find yourself staring at a search screen and realize "I need the parent's names!"
That's my 2 cents worth. :-) Igor
P.S. Just did a quick "rootsireland" query for James Murray 1796+/-3 yrs. Got 30 hits, many counties represented. Murray/Murry/Murra being the common varients.