There seems to be some confusion here. Green line is not a bus it is light rail, and yes, you need to buy a TAP card, which is $1. The paper ticket has been discontinued. However, you can keep the TAP card for your next visit to Los Angeles. The LAX shuttle bus to/from LAX to Aviation station is free. You can purchase the TAP card plus $1.50 one-way fare with cash or credit/debit card (I recommend a $5 day pass, which allows you to transfer to Blue line to reach Downtown LA or Long Beach instead of one-way fare).
http://thesource.metro.net/2012/08/0...s-to-tap-only/

If you want to take a bus to/from LAX, you need to take FlyAway Bus. You pay the driver when you board. No cash allowed... credit/debit card only.
http://www.lawa.org/welcome_lax.aspx?id=292