State Shuttle offers service between JFK and Newark airports, cost is around $30. If your final destination is Morristown then yes, take AirTrain from terminal to Jamaica Station, transfer to either LIRR or E subway to Penn Station and NJ Transit-Morristown line train. You may need to go from Penn Station to Newark-Broad St. station to catch certain trains to Morristown, check the NJ Transit website.
State Shuttle offers service