I would do a couple days in Kyoto (half day in Nara) if you can. Or an overnight at an onsen, we went to this one and it was a highlight for most people.

in tokyo:

http://www.shinjuku-robot.com/pc/ - this is a must! skip the bento it's terrible

http://www.ghibli-museum.jp/en/ - get tickets in advance, like today

Golden Gai - A tiny neighborhood of bars the size of a large closet. Some only serve 3 people, but it's so you interact and get to know each other. one of my favorite parts of the trip. We made friends with a local who got us in to some of the friendlier bars - some do not allow people who speak english.

Picture from our favorite bar. I don't remember the name, it was 4:00am and I was not sober at all

Karaoke - it's worth it, definitely hit it up at least once. I actually found the place we went because it had this amazing giant toast thing

I would say maybe try different accommodations. Our first night we stayed at a capsule hotel, then a hostel, then a business hotel, then a ryokan. All were really unique experiences in their own way. And if you can stay near the Yamanote line because it will get you just about everywhere you need to go.

Edit: Forgot a super important one! Learn the kanji for shochu. Do not get it confused with sake! Very different beasts.