IIRC, the lines that serve Summit don't go into NYC but rather Hoboken, NJ. That's o.k. as you can take a nice ferry ride from Hoboken to NYC. But if you get to Hoboken on a weekend there is no ferry from the train station. (Take a moment to admire the wonderful restoration of the 1934 Hoboken station.) You have to ride north through town to the 14th St. ferry dock. Take Frank Sinatra Way. It's a nice ride. You can also pass the bakery that was made famous by the reality show "Cake Boss."
Another option is to take the PATH train from Hoboken to the Word Trade Center stop or whatever they are calling it today. You may have to change PATH trains to do that. WTC is deep underground but there are elevators to get you to street level. PATH has some bicycle time restrictions as well.
In general, I would try to hit that area on a weekend. Less commotion.
Re: My response to your other thread, I have ridden Port Jervis to Philly in three days. Four weeks should be plenty of time unless you stay in places for a long time. Many years ago I meandered down the route from Bar Harbor, ME to Philly, with a detour to New Haven, CT, in about three weeks. If you think you may have some extra time and want to take a detour in NJ to see a wolf preserve, send me a PM. It wouldn't add any days, only expense (and some climbing). There is a campground next to the preserve and in the morning you can take a tour. Camping is $30 and the tour is $15, but I think you might get a small discount on the cost of the tour if you camp there. When I visited a few years ago the wolves started howling early in the morning.