Yes. One of the things that crews have to train for is the loss of communications with Earth. The astronauts also spend part of their time (when not in space, of course) working in Ground Control so they know both ends of the system. They could figure everything out on their own if they had to, with the exception of exact weather conditions on the ground.
Do these calculations have to be pumped out by a computer on board or could it be done by hand?
If its possible to be done by hand I’m completely in awe…
Orbital mechanics predates the space age by a good number of centuries. Having a computer onboard (which the shuttles and ISS do) would enable one to make a precision landing. The Apollo guys probably could have gotten themselves in the general area of where they needed to be.
In practice, though? I mean, exactly knowing where you are when you leave the ISS is a good thing to know, a couple degrees in one direction or another and you’ve got a major problem, I assume…
The crew of Apollo 13 managed to work out critical details with a sextant and a stop watch. They had a very narrow window of time in which to make one of their burns, or they’d have never made it home, and they managed to do it, so I’m sure someone leaving the ISS could do the same. (The folks on the ISS would presumably have an easier time of it, because they’d be able to pick from a wider variety of stars. Lovell and others on 13 had to use the sun, because they couldn’t tell what was debris field and what was a star.)