Have you considered an open jaw ticket, flying the reverse way? Fly into AYQ, fly out ASP. If that's feasible, Hertz has a promotion for unlimited kms, one way, no drop-off fee.

http://www.hertznt.com/promotions/default.htm

If not, I can tell you that I did do the trip without needing a car at all. Flew my parents out to ASP, where we stayed for 2 or 3 days. Caught the bus to Ayers; did the hotel-sponsored dawn bus trip, which got us out there well before sunrise. It takes you right to the base of Uluru afterwards, and it operates on a schedule after that. You catch any one back to the lodging area, whenever you want. We flew out AYR after a day there.

Other option - check with your company to see if you get special rates with one of the car rental places. IME the rates aren't that much better, and in some cases worse, but you usually get unlimited mileage.

Re: accommodation - reserve as soon as you know dates. I almost got frozen out; a last minute cancellation was the only reason we even had a place to stay.