If your flight lands at 13.30 you'll be doing well to be inside your Newark airport hotel room by 15.00 to drop your bags and freshen up. It takes ages to get to the courtesy bus point on the airport train let alone wait to be picked up by the hotel buses. Even a quick turnaround and you won't even get back to the airport rail station until 16.00.

You're going to be looking at well after 17.00 in New York City even if you want to pay a lot of money to get there on a "fast" train i.e. the type that makes Stephenson's Rocket look supersonic. Actually I'd put money on you not getting there until after 18.00 and if you've got a specific end point you want to get to on the Metro in the middle of the rush hour, well frankly by the time you get there it'll be time to turn around in time to get some sleep for your 06.20 check in time, which will need a 4.30am wake up call. Plus remember when you do arrive it'll feel like midnight.

Frankly, I think I'd settle for lazy dinner and a few beers in Newark and give up on NYC unless you want to hire a limo. The only thing you can do to improve things is perhaps find a Newark Penn State rail station hotel but it'll cost a lot more and won't help all that much.