While I haven't tried a single player game yet, I suggest the following: The player doesn't have a ship. This is the easiest to run, because you can just throw them right onto a planet/space station and let the game roll. Just ignore the 'your section of the ship' stuff, and ignore step IX of character creation. (If they absolutely want to take "Luxury", give them a luxurious apartment instead). Alternatively, if you want more starfaring, have them not be the owner of the ship, but rather an officer aboard, and have a few Crew (see page 35 of the 0.81 rules)
The big thing will be Factions. Since you only have one player, I suggest that you have the player create 2 factions, and that you as GM create 2 factions. This will give you enough tools to have some nice inter-faction dynamics.
Good luck! And please post your experiences, I'd be fascinated to see how a 1 player game of UW runs :D