I think it has to be soft sided, and not the hard type. You can easily bring a soft sided cooler in that can hold a drink and a sandwich with no problem.
It's best to just carry a drink into the park and not worry about the cooler. It's one less thing you have to carry. You'll have to put the cooler into a locker when going on the big rides like Hulk and the Mummy. They have free lockers near the rides, but it takes a few minutes to get the locker as other people will be doing the same.
If you really want to save money, leave the cooler in your car, and go back to the parking garage at lunch time. It's best to travel light in the parks.