No idea of the costs. Mears is the main limo company of Disney, with phones in the lobby of the hotels. Check their web site.
I'd suggest you head over to the HRH at 7am and check in. There is a chance your room may be ready. If not, the HRH will hold your luggage, give you room keys (FOTL), and you head off to the parks.
The other reason you want to check in early, is the line is shorter at 7am.
I tell most people to rent the car, depending on how long your stay is, and where else you want to go.