I don't know about the start times, but regarding the question in the subject line--the buses will drop you off at the Outpost, which is right in front of the trail ride corral (there is a small hut/building where you register, but it's pretty clear once you're there).
If you take an Uber/Lyft to Fort Wilderness, they will also drop you off in the same part of the resort, but my minimal experience with rideshares there is that drivers find it confusing (or at least ours--to pick us up at FW did--probably because of the busy entry roads for campers checking in?).
If you're wanting to do a pony ride, they are back at the new Tr-Circle-D barn at the opposite end of the resort (near the water), and you'll need to take an internal bus once you arrive at the Fort.