Realistically, you have very little or no time to spend in the city. Like others have said, take the subway and train from airport to airport. On a Saturday you can be stuck in bad traffic for a few hours. Don't forget, you have to check in an hour before your flight or they won't even let you check in.

Take the Airtrain followed by the E subway train to Manhattan and see how much time is left. Make sure to get the schedule for the NJ Transit train to Newark airport from Penn Station. Do not plan to take a car or bus to Newark from Manhattan.

MOMA is definitely out of the question. It is terribly crowded on Saturdays, they don't accept baggage at the coat check and if you need to leave your backpack at the coat check that alone can take up to 45min because the line will be very long. And then stand in line again to buy a ticket.