Hatta is beautiful and worth the trip but I am not so sure given the time restraints. I agree with Stimpy that if you are going to go to the desert then you should rent a car. Just a personal opinion. Again, driving there isn't hard and is very, very easy outside Dubai. Just stay out of the fast lane. No matter how fast you think youre going, there will be someone going twice as fast in a car worth more than I have made in my life. Al Ain is nice and you would be able to cross into Oman (no visa needed). Famous for its honey, some going for upwards of $100 a bottle. But the town is nice if a bit chaotic. Hatta is much quieter. In both Al Ain and Hatta are the old forts. Again, though, I am not so sure given the time constraints. I took a very roundabout way through Abu Dhabi to get to Al Ain and I made many stops before and after Hatta. Don't know the driving time straight from Dubai but I would think a couple of hours. If you don't have a hotel and go to Hatta, the Relaix & Chateau up there is beautiful and sometimes has a really good rate. I think it's called the Hatta Fort Hotel but I am old and senile and tend to forget things. In any event, I was able to get it for about $60-$70 for the night. It was one of my splurges. That would take you through the deserts and through some beautiful scenery.
One thing to consider is that *everything* is closed on Fridays until the evening. So if you are going on a Friday, prepare to get out of the city or go to the beaches.
Finally, a lot of the beaches, if not most of them, are private and not open to the public. Some allow you to pay to get on. I agree with Stimpy about shorts at the beaches. I was only referring to the cities. Although he is right about foreign men wearing shorts, the locals don't and when in Rome...
Here is a website with a few of my photos from the area so you have an idea of what it looks like. Hatta is in the east, Sharjah is in the West, and of course Dubai and Jumeirah have their own pages.
http://clam.rutgers.edu/~ddapas