Do you need to show anything at pick up to qualify for these rates?
Insider membership, but that's free to sign up at the Alamo site. The $35 coupon has been publicly posted everywhere. I believe it's an AA frequent flyer program code, so you could sign up for that too if you want to be safe. Seriously, I doubt they would ask for the coupon.
ETA: From Mousesavers:
American Airlines AAdvantage members (frequent flyer program, FREE to join) can get $35 off weekly rentals of any size car in the US, Latin America or Caribbean. Go to the Alamo website and use the coupon code AD2359FDF when booking. No physical coupon required. A Saturday night keep is required. Valid on rentals through 12/31/12. Thanks to Tom for info.