I will assume that your Archibald Stephenson was Church of Ireland. Records for the Church of Ireland parish near Eyrecourt are not available until 1871. If Archibald was Roman Catholic records are not available until 1884.
I looked at the Tithe records for the area and did not see a Stephenson.
I looked at the 1846 Slater's Directory for the town and still no Stephenson.
The 1855 Griffiths Valuation head of household listing for Donanaghta civil parish showed a J.B. Stephenson on Main St. in Eyrecourt but no other info. https://www.failteromhat.com/griffiths/galway/donanaghta.htm
I did not see a Stephenson in Donanaghta parish in the 1901 census.