So, you should call the hotel to ask, because I THINK, but I'm not sure, that they go 45 mins before magic morning at DL, Tuesday Thursday Saturday. So that would put you at security a half hour before. Personally I'd like to be through security and at the gate by then. So yeah, I think I'd uber.
They drop you at bay 13, in the shuttle dropoff area off Harbor. If you look at google maps, you can see this area where you walk into the gates, by the crosswalk across from Best Western Park Place Inn. It is the bay closest to the street on the north side of that island. It's very weird, because the bay numbers start with 1, on the south side, then run up to the top closest to security with bay 11 and 12, then a big empty spot, then 14... down to like bay 18 or 20, and then 13 out of order on the farthest end. No idea why.
It costs $6 per person, per day. You buy a pass from a kiosk that looks like a cash machine, in the lobby next to Starbucks. This gives you back and forth privileges all day. It takes off from right in front of the hotel; there is a little queue roped off to the right of the front door as you come out. Side note: both times we shuttled back at closing, they did not bother to check our passes or even look at them. We could have ridden back to the hotel for free. Not sure if it was because it was after the park closed and one of the last couple of runs of the night.