Charlotte,
Typically no, there are no discounted half-day ticket rates.
I always suggest that for the start of a vacation, a park for just a few hours may not be your best bet anyway. You are probably tired from travel, and you will only feel like you got too little time in the park.
I recommend that you plan a particularly fun dinner at one of the resort hotels. I like ones at hotels that have a fun or exotic theme. Some examples include the
Whispering Canyon Cafe in the Wilderness Lodge, the
Hoop Dee Doo Musical Review at Fort Wilderness, or
'Ohana or the
Spirit of Aloha Dinner Show at the Polynesian. But there are a ton of good ones to pick from.
Another choice is to go to Downtown Disney.
If you need some other ideas, just write me back and let me know what you like to eat & do.
Have a magical vacation!