You may find that during your first couple of days in Cancun, your main worry is dehydration. Water is plentiful, but will it be ridiculously expensive? (You might think so if the first water you see is on the price list of your hotel room's mini-bar.)
Bottled water is easy to come by for a reasonable price (approximately $1 US for bottles of varying sizes). Surprisingly, it is cheaper at some places (such as Chichen Itza) where you might expect to be overcharged.
Buy large containers of water at the convenience stores in the hotel zone and keep them in your hotel room. Even at room temperature they are refreshing. Drink a glass or two before going out, and a glass or two after coming back in -- even if you have just had a meal. Take your own water with you on any kind of long bus ride or any trip where you may be in the sun a lot.
All water in Cancun is purified ("agua purificado"), so you do not need to be concerned about drinking glasses of water in restaurants. When they ask in a restaurant if you want bottled water, I recommend asking for regular unbottled water.
You will also find Gatorade in the convenient stores. It is handy for replacing minerals lost through sweating, and is not very expensive.
If your stomach is feeling a bit queasy, try Sprite or 7-Up rather than a cola drink.
The chance of becoming sick or developing an upset stomach seems to depend more on how you handle the heat than on whether you drink the water. Some people become sick the first day and recover in a couple of days, while others who eat and drink the same things may not have any problems for an entire two-week stay.
Make sure to wear a hat, drink plenty of fluids starting before you start to feel thirsty, and if you feel worn out, queasy, or over-full during the day, take a 1-2 hour nap.
The temperature on a cloudy day will be much lower than if there are no clouds. Do not depend on a day to be cool just because it starts off cloudy. Bring your hat and sunglasses in case the clouds clear up during the day.
During the tour to Chichen Itza, you may be offered the opportunity to get a kind of bandanna containing a frozen gel pack, to cool you off during the tour. I was not able to understand whether this was free during the tour with an option to buy it afterwards, or whether you always had to buy it (for slightly less than $20 US). This is a clever idea, as Chichen Itza was the hottest place that I encountered in Mexico, and most places on the site had no cooling breeze. However, there are plenty of shady places to rest; I made it through the tour on a cloudless day and I don't usually handle the heat well. Of course, I did drink about 3 litres of water that day.