I have used Uber in the past but not in your area, but you can check for a quote on Uber.com. I was very pleased with them and considering you don't have much time, it will probably be quicker than a shuttle but cheaper than a taxi and easier. It's like have a personal car and you don't have to worry about money exchange because you prepay everything.
As far as 9/20, it will be a Saturday and as an FYI, locals like to flock to the park in the later hours so I would recommend you get there right when it opens at 9AM. There is Magic Morning hours that day at 8AM but it will still be less crowded then later on in the day.
There are already good suggestions regarding the rides you want to see so I ditto the recommendations. Maybe if time allows, try to see Haunted Mansion with the Holiday overlay, Ride in the front of the Monorail since you can't do that at WDW, go see the Marvel characters at Innoventions, and ride Small World because ours is a little different and has Disney characters in them. I know you're going to think our Castle is small but take the time to walk through it, that's always fun and the Sleeping Beauty dioramas are nice.
HAVE FUN!!!!

