You've only got a day, and it will take you a little while to get to the cruise ship terminal. Unlike what you might be used to in Chicago, the sights of LA are rather dispersed. With a day, the best thing would be to choose one area, stay there and see the sights there. The two best choices are:
Hollywood--Where the Walk of Fame is, historic movie theatres, live theater, many restaurants and bars, clubs--one of the liveliest parts of the city. Lots of hotels in your price range, like the Hollywood Orchid Suites. There might be deals on Hotwire if you're willing to go that route. Hollywood has a subway, but transit from LAX would require two buses, you may want to take a shuttle.
Santa Monica--Go to the beach, the Santa Monica Pier, the 3rd St. Promenade (a pedestrian only shopping street). Lots of nice restaurants. It's one of the most pleasant parts of LA and costs accordingly, but the Bayside Inn or the CalMar Suites might be in your price range. There's a direct bus (Big Blue Bus--Santa Monica's municipal bus line) from LAX to Santa Monica for $1. If you want to go to other areas, downtown Santa Monica is served by numerous bus routes.
What time are you getting in on Wednesday? If you leave the next day you have very limited options on what you can see. I don't know that I would stay in Hollywood just to see the Walk of Fame, you can do that in an hour or so and then your choices are pretty limited after that as to what to see. What other sites had you hoped to see?
How would you get to your hotel and the ship the next day?
Honestly I think I would just head down to Long Beach on the day you arrive to be close to the ship. from Long Beach you could take the Blue Line metro into downtown L.A. and transfer to the Red Line to get to Hollywood.
If you get in really early on Wednesday, you could stay in Santa Monica and use the Hop On Hop Off bus to hit the major tourist sites if you plan your time wisely.
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