Hey-
I work at Cornell and live close by. I usually end up taking a cab (about 25 bucks from LGA, about 40 to JFK). From LGA, u could also get on the M60 bus, get off at 125th street, and get on the 4/5/6 green line to 68th street/Hunter College. Walk East about 5 minutes from there and you are at Cornell (69th and York, on the East River). Back to JFK get on the 6 Green line at 68th street, take it south 'till in intersect the E train, take the E train towards Jamaica, and get off at the AirTrain stop...walk to to the stop, pay 5 bucks and take the hellishly slow monoral (airtrain) to your terminal! It can take forever to get to JFK that way, so I take a cab now....
good luck! cornell is a great place🙂