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If you decide to go out to JFK by subway, I would take the "1" train to 59th Street/Columbus Circle, and transfer there to a downtown "A" train. Be aware that "A" trains have different southern termini. Make sure that the train you board has a destination of Far Rockaway (Mott Avenue). These are the "A" trains that stop at Howard Beach/JFK station.
Ninety minutes sounds about right for the trip.

If you are at all uncomfortable about riding the "A" train through Brooklyn in the early morning, the LIRR from Penn Station is, as already mentioned, a reasonable option. Just stay on the "1" train to 34th Street, and connect there to the first Jamaica-bound LIRR train. The one-way off-peak fare is $5.25, if you purchase a ticket before boarding, and $10.00 if you buy the ticket on the train.

Whichever route you take, you certainly don't need to board the "1" train at 5:00 AM.