You have very little time.
Get your luggage to a hotel near JFK.
Then get the hotel's shuttle back to the airport to take the AIRTrain to Jamaica Station (NOT Howard BEach).
Take the E train at Jamaica Station to Manhattan OR take the Long Island Railroad at Jamaica Station. Both get you to Manhattan.
If you insist on staying in Manhattan, the E train is still good. There is a Sheraton hotel near the 53rd St./ 7th Avenue E train stop.
You have very little