I suppose you could take the Air Train to Jamaica, then take the LIRR to Penn Station. From there you could get NJ Transit or Path to Newark. I don’t know if that’s the cheapest, though.

My friend, take the Airtrain from the terminal to Jamaica Station. Please listen for the announcement, "Station D: Jamaica Station." The fare on the Airtrain, upon exit at Jamaica station is $5.

Once at Jamaica Station, transfer to the LIRR. Take any westbound LIRR train from Jamaica Station to Penn Station. A one way off peak ticket on the LIRR is $5.25, please purchase your ticket before boarding.

Once at Penn Station, transfer to NJ Transit. Take either the Northeast Corridor or North Jersey Coast Line from Penn Station to Newark-Liberty International Airport train station. There you can connect to the Newark Airtrain to the airport. A one way ticket on NJT is $15.50, which includes a $5.50 charge to ride the Airtrain from the train station to the airport.

I hope this information is very helpful.

Good luck

Take Airtrain JFK to Jamaica Station.
Get amount $8.75 by regular card, other amount. (It would say Airtrain/Subway.) $5 to exit system, and $2 will be for entry to subway downstair at Sutphin Blvd-Archer Av station and $1.75 for PATH train.
Take Manhattan-bound (E) train to World Trade Center.
Follow the temporality detour to WTC PATH entrance at W Broadway/Vesey St.
$1.75 will be deducted to equal $0 value.
Take newark train to Newark Penn Station.
Follow the bus lane sign to Raymond Plaza Bus Lanes.
Have $1.35 exact change (US Dollar bill accepted)
Take Perth Amboy-bound #62 or Toms River-bound #67 to Newark Airport’s A, B, C. Make sure to get off in airport, or otherwise, it’s additional fee.
New Jersey Transit bus operates on fare zone system.
You may also take Airtrain JFK to Howard Beach for Manhattan-bound A train to Chambers St/WTC station for PATH train as well.
Total Cost: $10.10

If you buy $4 MetroCard at HudsonNewstand.
Take $2 Q10 Limited bus (daily) Terminal 4 East to Kew Gardens-Union Turnpike station, last stop. If you’re not at Terminal 4, take Airtrain JFK heading to Terminal 4. As long as you don’t exit at Jamaica/Howard Beach, it’s free ride.
Transfer to Manhattan-bound (F) train to 34th St-Herald Sq.
Transfer to PATH train to Journal Sq.
Transfer free to Newark train to Newark.
Transfer #62/#67 bus.
Within two-hour, it’s free transfer so, you’ll have $.25 cent leftover.
MetroCard cannot be used on NJT Buses and there are no free transfer from subway/bus to PATH train.
Q10/F with 2-hours transfer/PATH: $3.75

By Airtrain JFK/LIRR/NJ Transit/Airtrain Newark
Take Airtrain JFK.
Look for Long Island Railroad ticket vending machine and purchase following ticket to NYC Penn Station.
Peak $7.25
Off-Peak $5.25
Weekend CityTicket $3.25
Buy that with $5 MetroCard printed on back. $5 to exit train and take inbound LIRR to NYC Penn Station.
From there follow the sign to NJ Transit concourse/waiting area.
At New Jersey Transit TVM, look for name and code next to Newark Airport.
It’s cost $15.00 one way to Newark Airport. Airtrain fee are encoded in one way fare.
Take either Northeast Corridor or North Jersey Coast Line with symbol EWR to Newark Airport.
Then take Airtrain Newark to your terminal.
Total cost:
LIRR Peak+Airtrain+NJT: $27.25
LIRR Off-Peak+Airtrain+NJT: $25.25
LIRR CityTicket Weekends On;y" $23.25