JetBlue flies pretty cheaply...
JetBlue flies pretty cheaply from JFK to Syracuse airport in one-hour long jet flights where you can take a cab into the city. Amtrak will take you from NYC up to Syracuse too which I have done many times. Getting to Syracuse from Ithaca or the southern tier pretty much requires a car, but Greyhound takes you north.
Within Syracuse there is virtually no public transportation except for Centro, the hard to catch bus system. If you stay up in Carrier Circle, you need a car or a cab. Best bet would be to rent a car from the airport, because you can't hail cabs and they'll be $20 each trip.

