It's not Japanese, at least, not exactly. In the year 2585 of the (little-used) Japanese Imperial Era calendar, Korea was a part of the Japanese Empire, and this medal is clearly from Korea — the first two characters of the horizontal line, reading from the right, say "Cho-sen", which is the name used at that time. In full, it says "Chosen taiiku kyokai" or "Korean Physical Education Cooperative Association". (The script is not nearly as contorted as some examples I've seen.) The other inscription is "Chosen Jingu kyogi taikai", or "Korea-shrine sports convention". The Chosen Jingu was the Imperial Shrine on Mount Namsan in Seoul (now demolished and replaced by a memorial to the man who assassinated Prince Ito), and is indicated by the "torii" gate and "shinmei-zukuri" roofs on the uninscribed side. It is very common for the premises of Shinto shrines to be used for athletic meets ; indeed, the Meiji Shrine in Tokyo was the principal site for the 1964 Olympics, and hosts the famous Koshien or high-school baseball tournament every year. This event was probably held in honour of the completion of the shrine in 1925.
I don't mind admitting, I like this piece, & kind of wish I had one.