I would do the NY airport bus from JFK to Grand Central Station. According to the schedule, the buses leave every 20 min from JFK from baggage claim and takes about 1.25-1.5 hrs to Grand Central. (first stop from JFK). The one way fare is $15 and RT is $27. Being "Mr. Frugal", I would still take the bus since the airtrain/subway combo is $7 and one has to schlep luggage for all the transfers. It's not worth saving the $8 to take the airtrain/subway combo considering it would take 2 hrs plus & some transfers. Since the Grand Hyatt-Grand Central is connected to the Grand Central Station, the walk shouldn't be far and I would guess the walk to be indoors.
Coming back, the subway/airtrain could make more sense if the Times Square subway stop is close. As an alternative, you could "cab" it to whatever station is the closest (Port Authority, Grand Station) and take the bus to the airport. I would say cab would be under $10 and bus $12-15. ($12 if 2nd half of RT, otherwise $15).
I did the reverse in 1997. I took subway into far Upper West Side when it was only $1.50 with free shuttle from JFK to Howard Beach subway stop. $1.50 vs. $13 by bus (or $35 taxi plus tax/tolls at the time), free shuttle/subway won out. On my return, I came into Penn station from seeing ex g/f from NJ and the Carey bus line was on strike from Penn station for that weekend. I took cab to Port Authority (about $6, worth it because it rained alot that day!) and took the Carey bus from Port Authority to JFK.
Bottom line:
1. Airport bus is fine for going from JFK-Grand Central and worth paying $15 vs. $7 airtrain/subway combo. Close to door/door w/o paying high taxi fare. Hyatt is attached to Grand Central.
2. Try cab to Grand Central/Port Authority, then bus to airport. My guess, max cost would be $25 ($10 cab, $15 bus) vs $45 or so by cab. Check with the hotel on what is most feasible taking time/cost into consideration. Take subway/airtrain if it would fit timewise and you don't have to walk far to the subway stop. There is a couple transfers in taking the subway/airtrain combo.
Maybe the NY based FTer's can do better than what I suggested. Have fun at the Freddies!
(I have to work this year--not much vacation time left
).