Is the 7-day JR pass worth it for my itinerary?

Hi,
It's almost covered by going to Osaka and back. You need a couple of side trips to break even - you could easily do it by seeing Himeji with Japan's finest castle for a day. Or go see Nikko for a day north of Tokyo. Or go visit Nara for a day, and do some sightseeing in Kyoto.
You can use the Pass on the Yamanote Line in Tokyo, as much as you want for as long as the pass is still valid. In Osaka you can also use it on the loop line.
Please note though that if you get the pass and activate it on May 30th, it would only be valid up to June 5th. Don't get caught in Osaka with an expired pass.
You could spend a lot of time in Tokyo and still not see everything. But a few of the best places to see are the Tsukiji Fish Market, plus the site that every last tourist goes to see, the Sensoji Temple. If you are in Tokyo on a Sunday afternoon, be sure to go visit Harajuku to see the youth with their wild fashions. Meiji Shrine and Yoyogi Park are next to it and well worth seeing also. And be sure not to miss the Shibuya Hachiko intersection especially on a weekend when it looks like half the city is crossing the street at once.
At night Osaka comes alive and has some great places to see, such as Dotonbori, plus the night views from the Umeda Sky Bldg, and Abeno Harukas Bldg are wonderful.

Best of luck.

Therefore, we should activate our JRpass on the 31st for it to be active on June 6, right? Where else can we activate the pass while in the city? Thanks for your tips. Can we also use the pass for a day trip to Mt. Fuji?

Hi,
Yes you are correct. You can see a list of where you can activate the pass here. You cannot use the pass all the way to Mt. Fuji. Most people either go to Hakone or the Fuji 5 Lakes area. Look into getting a Hakone Free Pass or a Fuji Hakone Pass.

Great! Thanks so much for your assistance. 😊