[masaegu]

Early June will almost always be better than mid or late June as far as the amount of rain we have in Tokyo. But since it differs greatly form one year to another, it's really hard to say if you should reschedule your trip. As samurai007 says, i don't think you will get so badly wet unless you plan on taking long strolls in Tokyo. A backpack will come in handy when you go shopping. By carrying some of the things you have bought in the backpack, you will have a better chance of having a free hand for the umbrella if and when it starts raining.