I and my family have been with AT&T all of our life, so far. My experience with them is outstanding. I have purchased my iPhones at times directly through AT&T online.
As the iPhone 5S is a recently new product, it will all depend on the number that they have at the time that you place the order online, and if there are others in front of you. For me, I did not do overnight, but I received it relatively quickly, within 3-5 days after it was processed.
There's no need to do overnight shipping. You won't get it any faster. Apple is listing 2-3 weeks for shipping, so AT&T should be similar. However, they overestimate their shipping times just to be safe. It will likely ship out in about one week after you order it.
You might get it faster if you order directly from a carrier. They *might* have them in stock somewhere in the U.S. If you order from Apple you're looking at maybe 2 weeks. I just got a gold 32GB model in just over 1 week. Ordered on the 23rd, received on the 1st. Supply constraints are definitely easing. Even if AT&T has to order it from Apple it wonk take any longer than ordering direct from Apple. They just pass the order straight through.
Sounds like they're ordering them from Apple just like if you ordered at an Apple Store. U don't know how AT&T handles it, but with an order direct from the mothership you can track via UPS every step of the way.