So, you have one full day and some partial days?
A trip to Mt Fuji would pretty much take your one free "full day". BUT, be warned - Mt Fuji is very shy and you may not see it. I think it's "hidden" more often than not so you may make the trek out to see it and end up disappointed. It's still a lovely day trip, as there are some nice areas in the Fuji Five Lakes vicinity, but it can be very difficult to *see* Mt Fuji in all of its glory.
Similarly, whether or not you'll see cherry blossoms really depends on the weather and luck. IF you happen to be there during the cherry blossoms, then Shinjuku Gyoen was really nice. Meguro River was also really nice, but extremely crowded. EXTREMELY!
Ueno Park had already blossomed and mostly fallen by the time we got there. And there was too much cement for my liking. I think we also visited the Imperial Palace moat, but we'd also missed peak blossoming time there.
You may need to keep some built-in flexibility to catch a bullet train north or south to view the cherry blossoms (if they're not in bloom in Tokyo when you're there).
Other than those aspects, the Studio Ghibli museum is really cool if that interests you (and if you're happy to book ahead). The aforementioned Tokyo-Edu museum, Asakusa / Sensoji temple, Tokyo Sky Tree, etc are all good. Odaiba can be fun. Harajuku can be fun for people watching and the nearby park is nice. Akihabara is also fun for a visit. I didn't rate Ginza too highly, but your mileage may vary.