The 12th Battalion Arkansas Sharpshooters didn't serve in the Army of Northern Virginia; they were part of the Army of the West (and its subsequent incarnations), the Department of Mississippi and East Louisiana, and finally the Trans-Mississippi Department.
There wasn't a significant number of Irish in the battalion. It was composed of volunteers and details from the 18th, 19th and 20th Arkansas Regiments, and the 8th Arkansas Battalion, and represented a typical cross-section of Arkansawyers.