1) Get on Airtrain heading to Jamaica Station
2) Put $8 on a Metro Card ($5 for Airtrain, $2.25 for Subway) and go thru turnstile and head to Subway.
3) Get on E-train towards Manhattan (WorldTradeCentr)
4) Get off at 34th Street/PennStation
5) Catch NJ Transit Train towards Princeton/Monmouth; ask at desk specifically for South Brunswick, NJ.
Expect this to take about two hours to complete.
How to go from JFK to Brunswick, New Jersey