I think that in general, people planning trips to Japan worry too much about the "rainy season" (tsuyu). No, it doesn't rain all day, every day. If you have really bad luck it will rain a lot of the time, though. So plan for it. One of the things that constantly surprises me is that people are always coming up with fixed, day-by-day itineraries that never seem to have any consideration for what to do about rain. So make sure you are prepared with proper footwear, suitable umbrellas (not too big, not too small, not too heavy), and a list of places that you can visit if it's raining. Bring light weather-resistant outwear, bearing in mind that even if it is raining it can still be pretty warm and you won't need a heavy jacket. And consider picking a hotel that is very close to a train or subway station that will be a convenient transit hub.
If you plan to be in Tokyo the whole time, then definitely research indoor activities. But also look at possibilities that involve train riding. If you were planning to take some long-distance day trips or overnight/multiple-night trips, then a Japan Rail Pass might already make sense. Add the flexibility it gives you to escape weather or at least be comfortable during it, and it could make it worthwhile to ignore the "calculator" approach that tells you whether you will "break even" or not. Or if the nationwide pass doesn't make sense for you, then think about some other possibilities. For example, the JR East Passes allow you to pick any five days during a two-week period, so you could possibly use some of those days for train riding in the rain (or possibly even to escape the rain--the farther north you go, the better your chances of that). Or the Tokyo Wide Pass might be worthwhile. Or maybe one or more of the local day passes. (I have fond memories of riding the Yurikamome line on a rainy day...)
Right before you leave, and regularly after you get there, check the weather forecasts and plan your days accordingly. Don't "waste" dry days doing something inside. On the other hand, as mentioned by rkold, the heat and humidity during that time period can be just as conducive to staying indoors as the rain.
If you like museums, consider getting a Grutto pass, especially if upon your arrival the forecast is bleak. Those passes usually pay off if you visit three or four museums, and they cover a few venues that aren't museums, such as Tokyo Sea Life Park. By the way, the Bridgestone Museum will finally be reopening in January after being closed for several years for renovations. It has a new name (Artizon), and I'm guessing it won't be covered by the 2020 Grutto pass (Bridgestone used to be on the list, and it was one of my favorites), but obviously you could just pay admission.
https://www.rekibun.or.jp/en/grutto/And yes, hydrangeas can be glorious in or after a rain. They will be better towards the beginning of your trip; by the end of June a lot of them are already played out.