A-1: My friend, the cheapest way to get from JFK to Princeton, NJ will be public transit. However, you could've landed in Newark, which is closer to Princeton and cheaper.
Any here's how to get from JFK to Princeton.
Step 1: take the Airtrain from the terminal to Jamaica Station. Please listen for the announcement "Station D: Jamaica Station."
Step 2: at Jamaica Station, purchase a $5 MetroCard from the MetroCard Vending Machine, and exit the Airtrain system into Jamaica Station. The $5 fare will be deducted from the MetroCard, when exiting the Airtrain system at Jamaica Station.
Step 3: Once thought turnstile (faregate), follow the directional signs to the Long Island Railroad, which is a commuter railroad. Purchase your LIRR ticket from the ticket machines on the pedestrian bridge overlooking the platform. A one way off peak ticket from Jamaica Station to Penn Station will cost you $6.25. If you're arriving on the weekends, you can purchase a CityTicket for $3.75.
Once you purchase your ticket, look at the display board on the pedestrian bridge for the next arriving train bound for Penn Station. It will tell what track the next westbound train will arrive on, which is usually tracks 1, 2 or 3.
Step 4: Once you arrive at Penn Station, exit the LIRR train an follow the directional signs to the NJ Transit Concourse. Purchase an NJ Transit ticket and take the Northeast Corridor Line from New York Penn Station to Princeton Junction, where a connection can be made to the Princeton Shuttle to Princeton.
Step 5: once you arrive at Princeton, you have the option to take a taxi or bus to your final destination in town, or if you have arranged a ride, you will be picked up at the train station.
The one way fare on NJ Transit, which is a commuter railroad like the LIRR, from New York Penn Station to Princeton via Princeton Junction will cost you $16.50.
I hope this information is very helpful.
A-2: The first answer is wrong. There is no subway at JFK.
There is an elevated AirTrain. First, take the AirTrain to Jamaica Station. Next, either:
a) Take the E subway to the Port Authority Bus Terminal and then take a bus to Princeton.
b) Take the LIRR to Penn Station and then take a train to Princeton.
c) Take the E subway to Penn Station and then take a train to Princeton.
A-3: Take the AirTrain to Jamaica. From there, take the Manhattan-bound (E) to 34th Street/Penn Station. From there, take the NJT Northeast Corridor Line ot Princeton Junction, and transfer to a "Dinky" to Princeton Junction.
You need a $7.25 MetroCard for the AirTrain & (E) train, and then $16.50 for the NJT train.