June in Japan is hot, humid, and entertainment spots are
crowded. The rainy season is continues into June and tapers off by the end of the month. Count yourself lucky if you don't have to commute by train during this time.
Breeding will be in full swing (first round long over in most places) and you're well into the culling will be underway depending on the variety. Koi farms will be devoid of foreign (and most Japanese as well) hobbyists during this time as there's not much to see except what's in the mud pond.
There's many festivals during this time, some depend on your locality. Tanabata is on July 7th, and you've got O-bon (you should know about that one being from Hawaii

) in August. Beer garden season is pretty much the whole summer, as are barbecue parties in parks (that allow them) or on river banks. Fireworks are a big thing this time of year, and you will see numerous displays throughout the summer. I think there's more fireworks displays in Japan than anywhere else in the world...it seems that way anyway.
Hope that helps.