The easiest way to do this is by taxi, either on the day by approaching one of the cabs outside the airport or by pre-arranging a transfer. We used Al Ghazal taxi's (most people seem to use them) and its a cheap (fixed rate of about 150 AED) 60-90 minute drive along the Sheikh Zayed Road highway and quite pleasant. Take a look
here for dummy bookings
and here for telephone numbers. We had our hotel arrange the transfer for us and i would assume yours would do the same, though its apparently very usual for visitors to walk up on the day and request the transfer to AUH.
Thanks,
Sarah