Yikes, that is coming up pretty soon! I'm not sure what you consider a cheap place to sleep, but there are plenty of hotels around the airport that would probably work. Earlier this year we spent a night at the Hyatt Place hotel by LAX, and it was affordable and a nicely done.
We were in L.A. over the summer for a few days, and here is what we did:
- Went to the Santa Monica Pier as soon as we left the airport.
- Visited Universal CityWalk for restaurants and shopping. You can also go to Universal Studios.
- Walked around UCLA.
- Visited the Getty Museum.
We stayed at the Hotel Angeleno on the west side of L.A., which is $222/night during your stay.
We also visited Disney, but if you are going to do that it would make a lot more sense to leave it for your week in Newport.