@ Suze - As Pella mentioned, you may want to hitch a ride with a fellow traveler living there, if the hotel will provide you the details. Otherwise, it'd be best to pay up for the resort's transport. You can check if the hotel has a shuttle to Abu Dhabi city (sometimes resorts provide such services) and then it might be easier, and cheaper, for you to go from the airport to the shuttle point (usually a known mall, corniche, heritage village, place of interest etc).
You can use the same option when you want to go to Dubai. Go to the Abu Dhabi city point, and then take a bus to Dubai. The RTA (Road & Transport Authority) bus cost AED 25 to Dubai (Euro 5). It's air-conditioned, and comfortable.
Regarding hotels in Dubai: Bur Dubai, Deira and Barsha are likely areas for cheaper hotels. You can also try to sign up for www.couchsurfing.com and search for a host in Dubai - which would mean that while the accomodation may not be as comfortable as a Hotel's, you'd have the advantage of having a resident as a flatmate, and zero accomodation expenses. I'm one of the Ambassadors for Dubai Couchsurfing, and would be glad to help you out if you were to visit Dubai.
Hope this helps! :)
Peace and Love :)