You don't say what time of the day you'll be there, but nowhere is more than 15-20 mins from the airport in a cab. Even with the surcharge for leaving the airport (AED20), it shouldn't cost more than AED60 into the city and AED40 back to the airport. AED100 = USD27.50.

Dubai Mall is next to the Burj Khalifa - 1200 stores, aquarium, food (and drink in the adjoining hotels). Or just get a cab to drive you around for a couple of hours. All cabs are metered.

Or the souks on either side of the creek plus a ride on an abra (ferry) between Deira and Bur Dubai for AED1.

It's still pretty warm and humid in Dubai though, if you're not used to it.

http://www.bigbustours.com/eng/dubai

hth