Where are you arriving from? If you're coming from somewhere else in the Eastern US you'll probably feel okay. if you're coming from Europe or Asia or even the West Coast of the US you could be pretty jet-lagged. Running around the city all night long might not be good for your overall health. If this is the beginning of the trip, you definitely do not want to extend yourself too much and be tired from the start.
This night tour https://www.coachusa.com/newyorksigh...productId=1383
starts before you'd even be at your hostel so it probably wouldn't make sense. And most others don't run at night. I like AliZ's idea of grabbing cabs back and worth. Walking is another great idea. You'll see, hear, and smell (all good and bad things) so much more on foot.
If you plan on taking a cab to the airport (I know they have fixed prices from Manhattan to JFK for $45 plus tip + tolls) you could ask the driver to go over the Brooklyn Bridge so you'd see that during the daytime (flight at 13:00 means getting to airport at 11:00 which means leaving Manhattan around 10:00).