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Here's the cheapest way to go, although it will take at least an hour. Take the AirTrain to Jamaica Station. You'll have to buy a MetroCard with at least $7 on it. They'll charge $5 for the transfer from AirTrain to Jamaica, and $2 for the subway. Take the E train towards Manhattan and get off at 42nd St (Port Authority)- it'll take about 45 mins. Transfer to the uptown 1 train and get off at 116th Street/Columbia U.

Good luck!