I presume Arthur Shaw was Church of Ireland? Records for the Church of Ireland parish of Cloone were destroyed in the 1922 Public Records Office fire in Dublin. I did not find any Tithe records for Cloone civil parish.
Based on your message, Julia was likely born around 1819. I searched on two different sites (Roots Ireland and www.irishgenealogy.ie) and found two possible leads but both records are later than 1819. There were no records around 1819.
Julia Horgan baptized August 24 1823 Mallow RC parish father John Horgan motherJohanna Sheehan
Julia Horgan baptized December 7 1824 Bandon RC parish father John Horgan mother Mary McCarthy