I'd suggest travellingto Cootehill via Armagh and Monaghan,stopping in Armagh to take in the sights if you have time. Then travel via Castleblayney, Dundalk, and Newry and towards Newcastle. There are several scenic deviations on this leg, if you have time.
Belfast to Newcastle via the A24 or A2. After Newcastle you have a choice of routes, I would pick the scenic coastal route down to Kilkeel and Newry. The alternative mountain route is narrow and hazardous in places. I drove the route from Newry to Armagh once and swore never again. Better to go to Dundalk and follow the route to Cassleblayney or Carrickmackross. Roads to Cootehill are twisty and narrow in places. I would allow three hours between Newcastle and Cootehill.