Arriving NYC 4pm, departing 9am; is a Statue of Liberty visit possible?

I'm afraid you will not be able to see the statue of Liberty up-close, as the ferries only run during the day (last boat leaves at 5pm). I'd recommend perhaps taking a stroll around Times Square and perhaps taking in Central Park. One way of doing this is below (assuming you have fairly limited baggage, as this would prevent you from doing most things without having somewhere to leave it):

1. Upon arrival at JFK, catch the Green AirTrain service to Howard Beach, and there obtain a Pay-Per-Ride MetroCard from a vending machine. You will need $10.50 credit which costs $11 due to the 5% bonus for topups and a $1 card fee
2. Change for the A train subway service toward Manhattan. Alight at 42 Street and wander across to Times Square, then head northwards and spend some time in Central Park.
3. Catch the C train from any of the Subway stations along the western edge of the park to 34 Street. Change there for the NJ Transit train for Newark Airport (you'll need to buy a seperate ticket for $13, the MetroCard is not valid). I suggest you spend the night at or near Newark due to your early flight and cheap accomadation found there.

I've avoided any complicated money-saving options (such as getting the bus to or from the airports), as since this appears to be your first time in NYC and you have limited time available.

If you really want to see the Statue of Liberty up close, Circle Line Sightseeing Cruises has an evening statue cruise that departs daily at 7:30 PM from the South St. Seaport. The fare is $31 as of this writing, and the cruise lasts one hour. However, the boat does not appear to land on Liberty Island.

Bask in the glowing panoramic view of NYC’s celebrated towers, memorials, and of course, glorious Statue of Liberty! Narrated by informative and entertaining guides, this evening tour aboard NYC's most iconic boat is an unforgettable experience.

Things to Know:

• 1-hour evening cruise along the East River
• Spectacular views of the Manhattan skyline, Governor's Island, Brooklyn Bridge and the Statue of Liberty
• Enjoy engaging and informative narration by our guides

(I have no affiliation with the above operator; I just Googled "evening statue of liberty cruise". I was unable to find any other operators offering evening cruises of this type.)