I'm 99% positive there is a shuttle bus from LAX to the El Segundo green line station (board the bus outside baggage claim at the terminal you arrive at and keep an eye out for a sign on the bus that says "Green Line").

Yes you can get to downtown this way but it might take a LONG time. You'd have to take the bus to the green line station and take that to the blue line, then transfer to the blue line and take that to downtown. Not sure which station you exit to get to the Westin Badadventure.

Check out this website to see if it helps any more:

http://www.mta.net/

Here's another link:

http://www.lawa.org/lax/laxframe.html

Click the Ground Transport button, then Public Transport button for more info.

Enjoy your stay here in Los Angeles!