You would need a 00-90 screw to fit that hole. The problem is that the longest I see is 1/2" and the base of that lamp is 1/4" thick. If you broach the hole a little bit, you could use an 0-80 screw which you can up to 1" long.
You might be able to find small wood screws such as #0 in Hobby Lobby, or similar store. They may also have some alternative fasteners.
For nails you would need something less than a 1d or something like Micromark sells as track nails but you may have to file one side of the head to fit the base of the lamp.
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