You purchase your Airtrain and subway fare on a "pay-per-ride" Metrocard from a self-service machine when you exit Airtrain at Jamaica Station. There are nice folks there to help. If you purchase your Metrocard for $8 or more you'll get a discount so if you expect to use the subway/bus/return Airtrain during your stay this is the best deal. Take E subway to 53rd St./Lexington Ave. and transfer to the downtown #6 subway one stop to 42nd St./Grand Central. Walk east on 42nd St. about 3-1/2 blocks (these are long blocks, especially with luggage!) to your location between 1st and 2nd Aves., or you can take the M42 bus (runs along 42nd St.) eastbound from the subway to your location, use your Metrocard for the bus fare. Subway is ok at this time of evening.
You purchase your Airtrain