Just your basic Arrow staple gun should work fine. On any of them make sure you have the gun tight against the wood, this will help seat the staples and prevent jamming.
A buddy of mine has this cabinet he wants to give away. I want it so bad, but just don't have the floor space for it. He says if I don't take it, it's going to the dump. Each set of 3 drawers is removable and have no "back" to them.