First trip to Japan: 15-days during rainy season

So, I've been looking forward to going to Japan for a good portion of my life and initially when planning the trip, we had a horrible travel agent. We originally wanted to go mid, late-May, but long story short, the travel agent screwed up and our trip got pushed into June as a result. Now all I'm reading about is how terrible of a time June is to visit Japan. I'm really hoping the rain doesn't ruin our trip (ironically, the same thing happened on our honeymoon).

So, I'm very much curious just how bad the rain actually is, and if it'll ruin the whole trip. I'm leaving in just over a week right now.

5 Days in Tokyo

Day 1: Nothing planned for the first day, since we assume we'll be dead from travel, and just want to explore what we can.

Day 2: Tour guide of Tokyo from 12-4, planning to explore Akihabara for the rest of the day, as we have reservations at the Eorzean Cafe that night.

Day 3: Explore Asakusa and Harajuku

Day 4: Explore Yokohama, TeamLab Museum, Ramen Museum, maybe Chinatown

Day 5: Any last minute things we wanted to double back and see before getting on the train to Kyoto. I am a bit curious about the trains though. Can I buy a green car seat at the counter/kiosk? Or do those need to be booked in advance? Thinking it might be worth it for having a bit more space for luggage and stuff.

6 Days in Kyoto

Day 1: Arrive from Tokyo, settle in, and aimlessly explore

Day 2: Half-day tour from 11:30-3:30, no other plans. More exploring. Maybe Fushimi Inari Shrine, or some others?

Day 3: Go to Uji for Tea Tour. Maybe the fire ramen restaurant at night?

Day 4: Kyodo Class in the morning, Ramen Cooking class in the afternoon, Toei Kyoto Park after

Day 5: Bushido Class in the morning, explore Kinkakuji temple, and arashiyama bamboo grove after. Followed by an Izakaya cooking class

Day 6: Anything we stumbled upon we want to revisit before moving to Osaka.

4 Days in Osaka

Day 1: Arrive from Kyoto. Food tour in late afternoon, maybe Aquarium if it's rainy.

Day 2: Half-day tour with guide, explore Tenjinbashi-suji shopping street

Day 3: Day Trip to Nara for Deer Park, Isuien Garden, Yoshikien Garden, Kasuga Taisha Shrine

Day 4: Explore whatever else caught our eye before moving to Fukuoka.

3 Days in Fukuoka

Day 1: Arrive from Osaka. Toto Museum, explore Yatai Stalls

Day 2: Explore canal city, and half-day guided tour

Day 3: Early morning exploration before going home

As you might be able to tell, we're not really big on planning, and usually extemporaneously just like to explore places when we get there. But right now, my biggest concerns are how much luggage to actually bring for the trip (maybe I just do laundry somewhere halfway through), how much the rainy season will dampen (heh) the trip (is it a full day rain, or something unintrusive like 15 minutes of light rain every couple of hours?), and whether or not buying a Kansai Area Pass is worth it? (Doing the math, the JR rail pass didn't make sense, but 9 days in Kyoto/Osaka leads me to think that might work out)

Thanks all.