I don't think there are any direct flights from Cork to Newcastle - cheap or otherwise
2 choices basically
Take an indirect flight. Aer Arran to Dublin and then on to Newcastle with Ryanair would probalby be most cost effective option here - otherwise via London
or
Fly from Cork to somewhere you can easily get the train to Newcastle. Manchester is probably the best bet - "BMIBaby" fly relatively cheaply Cork -Manchester and there's a train staion right in Manchester airport and it's about 2 1/2-3 hrs to Newcastle.... alternatively Aer Arann fly Cork to Endinburgh which is a shorter train journey (but you have to get from the airport to the station) ... Also BMI fly to Leeds Bradford - not sure about trains from there ?