Well, that IS the game, honestly. Once you manage to "solve" the ship and money problem, it's "easy".
Of course in Star Trek, the characters come from a position of power (Kirk is the Captain, after all), it's just far easier for him to lose that power (i.e. get killed) and he doesn't have the autonomy that he's used too.
Depending on the state of the society, being undocumented may or may not be a problem. Do they land in 1960's Mephis, or 1960's East Berlin -- different problems entirely.
If they can solve the documentation issue, then the rest is, as they say, "just work" unless they keep running up against the authorities. (Imperium with a bit of Paranoia, perhaps?)