If you're getting there right after lunch, I would suggest not wasting a day on your pass unless you are getting a length of stay one and it won't matter. I agree that I would go nuts at the pool, but I see their point.
I would suggest getting settled, checking the hotel out (depending on where you're staying, that in itself can be an adventure!), then head to Downtown Disney. You can feel Disney magic there, its free parking and entertainment, hit the shops, have dinner there, see West Side, possibly do DisneyQuest, etc. Then you don't feel like you are wasting money on a park ticket if you do decide you're really tired and want to head back.
You can also grab a movie or something down there if you want down time but dont want to sit in the room.
This time we're arriving around dinner, so we plan to check in, get settled, then head to the Boardwalk. ORder pizza at Spoodles and relax out there watching the people go by. Then the next morning, we're hitting the parks hard (and starting the day with the Princess breakfast!)
HAVE FUN!