You can take the A Train to far Rockaway exit at the JFK Howard beach station transfer to take the Air Train that will take you to the airport
Get the E train direction uptown Jamaica. Get off the E train at Stuphin Blvd. There you may get the Air train that stops to every terminal of John Kennedy Airport
Take the LIRR or the E train to Jamaica Station for LIRR. You disembark at Archer Street for the E train. Go upstairs and take the train to the plane. It stops at all terminals. Hope that helps.
You may transfer to the E train on the Uptown side to Jamaica Queens. Exit on Stuphin Blvd. Here you can get the Air train that will take you to all the terminals of JFK Airport.
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