We drove up into the mountains to see some of the local forts, we also drove ourselves into Muttrah to go to the souks (real souks, not dubai style heritage souks...!), and we drove down the coast a bit to beyond where the Al Bustan is, where the scenery is amazing and they have lovely beaches.... great diving and snorkelling there. Also nice to have a bit of freedom, just driving around. They do have some of the same shopping centres as Dubai and AD (smaller scale City Centre, and a Lulu Centre, which I love, all those fab Indian silks etc....). We also investigated some of the other hotels for future reference, definitely were happy to be in the Chedi. The Al Bustan has a beautiful beach, but is much bigger than the Chedi, full of brits and their children, which is fine, but we were happy in the slightly smaller, more cosmopolitan, less children (but good for v. young children and babies) Chedi. I live in London, so although everyone says that the Chedi is £££, it all seemed pretty reasonable to me (!), and certainly having eaten a few meals in the other hotels (Al Bustan, Hyatt (don't go there whatever, it's like a bad hotel in Vegas, not even kitschy-cool!) and the Inter Con), all the prices are much the same. I was pg when I was there, had the odd glass of prosecco in the evening, so not boozing much... I would guess the prices are much the same as Dubai, though I haven't been there for about 5 years.....
How long would you be going for? We stayed for 12 days, so definitely enjoyed being out of the hotel zone for 5 days with the car in the middle of it all, but if you are going for a 5-7 day trip, you might decide to do your moving around by taxi.... car hire wasn't expensive though, and we arranged it through the hotel....
Hope that helps!