(1) Here's a link to the map of the Valley, with the Wilderness Center marked:
On this map, the post office is to the left of the WC, and left of that is the Visitors Center.
(2) Here's a link to another transportation map:
On this 2d map, the backpacker's campground is directly under the "o" in the word "North" of North Pines. You can follow the paved (red-dotted) path from the backpacker's campground to the Wilderness Center, even at night. It's about a 10-15 minute walk. The number 2 in the green circle is the grocery store. You can sit outside the wilderness center and I think you'd be fine if you got there by 4 AM. Remember that you are getting a permit to start the day AFTER you pick up the permit, not that same day. So if you are starting the JMT on Friday, you need to pick up the permit on Thursday.
Edit: the museum has a movie theater showing a history of the valley thats air-conditioned, the grocery store is air-conditioned and the Awahnee has some reading rooms in the back where you could probably hang out for a few hours.