Greyhound from Port Authority Bus Terminal goes to Albany ($31), departures on 10/8 at 9pm, 11pm and 11:30pm. Return to NY on 10/12 leaving Albany at 4:25am arriving NYC 7:15am, you'd be at JFK around 8:30am. Did not see a Stuyvesant stop on the route. Amtrak's Empire train to Albany ($37) from Penn Station (also stops at Hudson ($32) south of Stuyvesant) departs on 10/8 at 8:50pm or 11:50pm. Return to NY on 10/12 leaving Albany at 4:50am or Hudson at 5:15am arriving NYC at 7:10am. There are NYAS express buses to/from JFK and Penn Station or Port Authority from each terminal or E subway from either location to Jamaica and Airtrain to terminal. As you'll be traveling in the wee hours of the morning I'd suggest the train to Hudson or Albany and your friends can meet you there.
Greyhound from Port