First Trip to Japan in June

Hi,
First, you write you are arriving in Osaka on June 8 and leaving out of Narita on June 6??

At any rate, if you are leaving out of Narita and not returning to Osaka, a 7 day pass would only pay off by taking a side trip to Hiroshima/Miyajima, as you suggested.

In the Tokyo area, taking a day trip to Nikko also makes a great day trip. Near Kyoto, seeing Himeji is also very nice, with Japan's finest castle.

Virtually anywhere in Tokyo is close to a train or subway station. If you want the bright lights and neon though, Shibuya and Shinjuku are 2 of the best places though. For hotels and souvenirs, a lot depends on your budget and interests.
Regarding seeing Mt. Fuji, I recommend taking a day trip to either Hakone or the Fuji 5 Lakes area. A JR Pass can get you there part way, but there are regional passes for both.
In Tokyo 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.
There are also many beautiful gardens if you like, such as Shinjuku Gyoen, Hamarikyu, Rikugien, Kiyosumi, and the Koishikawa Korakuen Garden.

Best of luck.

Thank you thank you! Sorry for error in typing. Arriving Osaka on June 7th and leaving Narita on June 16.