Spray cold water over the entirety of your shed with a garden hose. This will help cool the physical structure.
Keep the door to the shed open all day. If you have a problem with bugs and bees, hang a mosquito net over the doorway.
Give your shed a window if it doesn't already have one. Keep the window open and either place a screen in it or use a mosquito net over the opening to keep out any bugs or other intruders.
Plant a shade tree on the side of your shed that is exposed to the most sunlight, if the shed is already standing. If you are constructing a new shed, be sure to build underneath a shade tree or in a shaded area.
Paint the shed a color that will not reflect or trap the sunlight such as white, light gray or light blue.
Cover your shed with a climbing vine such as bougainvillea or ivy. Place poles spaced two feet apart around the entirety of the shed except the doorway to help hold up the vines.
Install small awnings over the door and window.
Install vents in the shed's roof and floor to provide more air movement.
Place a small air conditioning unit in the window of the shed if it is going to be occupied regularly. Cooling a small shed takes less energy than cooling an entire house.