Many people are surprised to know that Japan summer is quite hot, this is probably not the image they have of Japan.
In my opinion, the weather start to be quite unpleasant from June, when the rainy season is coming. From June, temperatures are not especially high, but the constant humidity in the air makes you feel kind of heavy, and sweaty all the time.
I will never get used to Japanese ladies who even at 37 degrees, wear the total look : long sleeve + opaque tights + mask + hat with huge visor. I still didn't figure out how they could survive under so many clothes.
Same with mens in suits in summer!
After living in Japan, all my foreign friends changed 2 habits that they would never do before coming here:
1・ Always bring a small towel with you
Those smalls towels you could bring everywhere with you are a must in the summer. Once you will finally accept that yes you will sweat all the summer, you will find the usefulness of those towels and will use it everytime you're outside (subway, streets, before meeting a client...).
2・ Put a thin t-shirt under your shirt
I have several times heard that putting a thin t-shirt under your shirt was kind of "old fashion" and didn't looked good. Here in japan you will understand that this is more than useful. It is no hotter than only the business shirt, and there is a real meaning to those t-shirts you could find for cheap in Uniqulo-GU and many other shops. Those T-shirt are real sponges ! and the difference with and without it is quite impressive.
Another important point : drink a lot ! This is a real necessity. You could find in convenient stores "lemon+salt" drinks and candy especially sold for fighting heatstroke.
The last recommendation for the most heat sensitive : the usb fan to install on your desk!