My friend, the cheapest way to get from LGA to NYU is by public transit.
Step 1: purchase a $4.50 pay per ride (regular) MetroCard or a 7-day unlimited ride MetroCard for $29 from any Hudson News stand at the airport or from the MetroCard Vending Machines located within the airport terminals, and take the Q33 bus from the Main Terminal to the 74th Street/Roosevelt Avenue subway station. If you're coming from the Marine Air Terminal, please take the Q47 bus to the 74th Street/Roosevelt Avenue subway station.
Please note: The MetroCard Vending Machines at LGA only accepts credit and debit cards. If you preferred not to use your credit or debit card at the airport, there is a change machine right next to the MetroCard Vending Machines that will change dollars bills to coins for the bus.
Step 2: at the 74th Street/Roosevelt Avenue subway station, transfer to the "E" subway line (Blue Line on the map). Take the Manhattan-bound (westbound) train from Roosevelt Avenue to West 4th Street-Washington Square, which is the closest subway station to NYU.
The walk from the West 4th Street-Washington Square subway station to main campus of NYU take about 1 to 2 minutes.
I hope this information is very helpful. If you have any questions or need travel directions while in town, just drop me a line by clicking on the avatar and e-mail icon in the profile. I'll be glad to help.
Native New Yorker