The surname McCourt is found in all the provinces in Ireland but especially Ulster, particularly in Counties Antrim, Armagh and Monaghan.
McCourt, in Gaelic Mac Cuarta or Mac Cuairt, was an Oriel sept, which was based in southwest Armagh. The ancient territory of Oriel included Counties Armagh and Monaghan and parts of south Down, Louth and Fermanagh. The parish of Cappagh, near Dungannon, County Tyrone was originally in Gaelic Ceapach Mhic Cuarta.
A good starting point for records of Counties Armagh and Down is RootsIreland at
https://www.rootsireland.ie/