Hi Steve,
Sorry, I cannot help with parking/shuttle service. However, I have done the S to N day hike, and really enjoyed it. I departed on the South Kaibab trail and exited the North Kaibab trail. When I did it my sister dropped us of at the trail head, and drove around to the North side. We stayed the night (after the hike) at the Grand Canyon Lodge. If you can get a reservation you should do it. Magnificant lodge, and not many other choices. The South rim has several hotels/motels nearby. If you stay the night prior to the hike they might even drop you off at the trail head.
If you cannot get a room at the Grand Canyon Lodge you "might" have the energy to drive back to the South Rim, but it is a long drive.
My number one lesson from the hike was learning the importance of fueling the body. I did not take enough to eat, and really had to work hard to hike up the North Kaibab trail. Not like Whitney, where gravity will get you home. I would advise hiking on Friday. That way you would have Saterday to drive Julius's car back to the South Rim. Both of you will likely prefer departing to home from the south Rim.
Just my 2 cents:-)

Enjoy!