Manhattan's actually going to screw over a lot of natives because it was, at that point, a massive hunting ground that supplied the tribes with food. Then again as noted, it's fairly likely the natives are going to be hit hard with a few plagues from the upstreamers anyway.
Manhattan has a self-contained power grid but its plants are gas turbines, not sure how long they'll be able to operate but probably days, no more than weeks. Long enough to get some radio messages to the Pentagon and link up, though.
There's enough tooling and machine shops for them to kickstart a bit of industrial revolution, and between Manhattan and the Pentagon there's going to be plenty of people with the knowledge base to MacGuyver up some light industry. I would estimate they wind up decaying to early steam technology before starting to advance again.
They should have enough boats to fish, and game in that area is very plentiful so they will have a decent chance of staving off initial starvation. The population will have to rapidly leave Manhattan and spread out to claim enough territory to feed themselves, however. They will know where all manner of mineral deposits are and the Appalachians are basically made of coal so initial resources won't be terrible. New York actually has a large number of mineral, gas, and oil deposits historically, it was so heavily settled for a reason and it wasn't because the UN set its headquarters there in the early 1400s.
Animals are going to be a major bottleneck, in the sense that the Americas have no domesticable ones. If they're extremely lucky, there might be a cargo ship with some livestock nearby. There are some mounted police and, as far as I know, a single dude ranch that will go downstream. So they will have horses but I don't know if any cattle will come with. Possibly they'll get some chickens if fortunate. Dogs and cats will be plentiful. If they get cattle and chickens they'll be well set for future farming and ranching operations. There are going to be at least some heirloom vegetables in hipster bistros if nowhere else, and seeds for flowerpot gardens and those are going to be gold for getting early farming working, those seeds will be more valuable than a penthouse apartment at that point.
There are enough subways in Manhattan that they might be able to scavenge all that rail and use it to build a railroad to some nearby resource deposits. This will take some doing, it's going to be backbreaking work clearing the path and even worse building a working locomotive but it's at least possible. Reaching the Cumberland gap is probably beyond them so they're going to be stuck east coast-only for a long time, but the east coast also has plenty going on to support them.
Linkups to the Pentagon are going to be best accomplished by going most of the way by boat. They should have enough hobbyist sailors to get some sailing ships working at the beginning before they're able to build a steam engine and get a supply of coal for them. I'm not sure if the Pentagon comes with enough surrounding land to have any boats available but it's likely to have plenty of engineers and sailors of its own, and at least some of them will know how to build and sail old-school craft.
The big question in my mind is firstly if they'll be wiped out by the native tribes. At this point the density of natives is going to be around 50-100 times greater than it was post-smallpox. The Pentagon will have enough defenses to survive, and isn't on critically valuable land anyway, but Manhattan is not so well defended and also happens to have just displaced so majorly important hunting grounds so is more likely to be attacked.
Secondarily there's some question as to whether they'll simply be absorbed by the native tribes. I see this as a touch more likely (even just police weaponry is going to be horrific in repelling initial attacks) and it leads to interesting places. If they have any cattle, that's going to spread cowpox to the population and that grants immunity to smallpox. The population of Manhattan and the Pentagon will, themselves, have a number of doctors and a knowledge base of which diseases will spread along with how to combat them. With 400 years to play with, it's likely that the assimilated upstreamers will have spread enough knowledge for the natives to at least have a well developed steam-based industry, and it will include enough white guilt and historical information for them to know what the Europeans did to the native population. Odds are they'll be sending ships to Europe no less than a century earlier than Europe was able to reach the new world, and the conquest may well go the other way around.