I went to high school in such a building in Valdosta, Georgia. It really isn't too bad. High ceilings and cross ventilation help a*lot*, and due to thermal inertia, the day and night temperatures get averaged, and the daytime temperature indoors is a good bit cooler than outdoors in the middle of the day.
What's *bad* is "modern" (1960s) schools and other "modern" buildings that were built without air conditioning. Somehow the architects just forgot that they were building a kind of architecture that only worked with air conditioning. The worst was a dormitory I lived in at Yale (Helen Hadley Hall) whose central HVAC system was deleted from the plans at the last minute. The place was hot for half of the year and smelled bad all the time.
To this day the Executive Office Building has only window units -- I noticed while walking by it recently.