Here's another vote for avoiding Ramadan. Food options are very limited during this time.
As well as the other suggestions already made by people, I would go to the Dubai Creek (you can use the subway) and take a ride across in the traditional abra (boat) to the spice market and gold souk. Great street foods, interesting sights, you might want to buy something. The abra ride is very cheap and it's cooling to be on the water. Eat some local foods - not just in the hotels. Go to one of the beach resorts and have a drink or coffee overlooking the beach. Swim if you want to, or try some of the adventure sports like hoverboarding/flyboarding. A hobby of mine is to visit the supermarkets and check out the different foods. The date aisle will stretch for miles and who knew that there were 20 or more different types of dates! It's also fun to go out to a shopping mall and just people-watch the local families enjoying their time together.
If you have a car, take a drive to Fujairah (about 90 mins). The scenery on the trip is amazing, and the town is coastal (Arabic style) and interesting to look around. There is a beautiful mosque there.