Oh Dear God! A points stay???????????? You'd be better off at the Century Plaza or using priceline for an airport property and avoiding this place.
At any rate, I will help you out with the Green Line. I was supposed to stand by at 7 a.m. for a flight to ORD. I left the hotel shortly after 5. The green line station (an elevated rail system similar to Chicago) is literally next door to the hotel. You walk out of the hotel and turn right. Walk up a sidewalk to the entrance. It's a 2-3 minute walk. I did see a sign that said one sidewalk would be closed beginning in October, so YMMV on how to get there. But it is next door.
You get on the Green Line (left track if you're facing the hotel) away from the Beach (Redando I believe). Get off at Aviation. It's 3 stops. Then, you have to wait on the G bus to take you to the airport. The train ride is fairly quick. It took 10 minutes at the most. Waiting on the G bus can take forever, especially on weekends. In sum, it took over an hour to get to the airport and I had to stand by on the 8:10 instead of the 7 because of how much time it took.
Mind you, if you have the time, $1.25 is a great deal to the airport (which is between 3.5 and 6 miles away depending on which hotel employee you ask). Taxi would be at least $15 (which is also what Super Shuttle wanted). But know that it can take over an hour to get to the airport and plan accordingly.
The G bus is the same bus you can take to change terminals. It will stop at all of them. It's a bit slow but easy to use.
All in all, the LA transit system is easy to use. The stop nearest the hotel is Douglas Street. There will be 2 more stops and then Aviation. Then, look for the black and white signs that say LAX Shuttle or something to that effect.
However, if you don't have the patience for that and you're wanting a budget airport hotel, try priceline. It will be $40-$50 well spent. I know if I had it to do over again then I'd stay at the Century Plaza which is about 7 miles from the airport and use super shuttle or else go Hilton, Marriott, or Starwood for the LAX area.
Hope this helped.