Originally Posted by csdf

1) You could get dropped off at the Shangri-La, which is opposite the Grand Mosque. You could go for a wander through the Souk al Bahar which is attached to the hotel, then grab a cab over to the mosque. Look online for timings - though you can turn up any time (except on a Friday) they run free informational tours at certain times which are well worth trying to catch. The Shang is 15-20 minutes from the airport.
2) Pretty much any hotel will do it. Even if you end up on the Corniche you'll only be 35 minutes from the airport.

If you want to wander, I would suggest a walk along the Corniche. The Western end has the Emirates Palace and the Jumeirah Etihad Towers, the Eastern end has the harbour, the fish market and the carpet souk. You can even hire a bike if you like and cycle the length (it's quite long, maybe 15km).