I didn't read the whole thread but here are my suggestions based on my times in japan and my family living there.
I suggest going to Shibuya while in Tokyo, which not only has one of the worlds (i think it is the biggest) crosswalks on it, but the Hachiko statue which is right outside the main subway station and is pretty easy to find since people congregate around it after they get out school and work. In the same area several blocks away there's also the Salt and Tobacco museum, which is relatively small but I thought it was a neat little place. Shibuya is also a fun shopping district with a giant Tokyu Hands store and a lot of pachiko parlors and a bar district (its really just a tiny street) that each bar can fit about 5-10 people and looks out of the Edo period. It's called Nonbei Yokocho and is literally right next to Shibuya station and kind of hidden. THere's also a bar owned by a dog somewhere but I forget which one it was.
If you're in the Roppongi district, not far from (I forget which line it is) there's a little american ran bar there because the US embassy is nearby. It's called Geronimos and the staff are really friendly there if you want to ask about stuff going on locally. As a side note if you're american, you can tell the embassy online when you're planning on being in japan so they know you're there in case you get in trouble. Very helpful. The embassy compound is also nearby in roppongi which is hilarious because if you walk past the entrance, the entrance guard is usually japanese and assumes you live there and are lost so they try to flag you in. (As a side note if you do get inside, they have a freaking gorgeous koi pond)
I also suggest Harajuku, which is like the go-to street clothes shopping district (the street is called Takeshita and has a daiso, which is essentially a japanese dollar store which is great for gifts to bring home to friends), home to the bridge where all the lolita kids hang out on Sundays, and is also right beside a huge (i think shinto) shrine and park, which is beautiful especially when the tulip garden is in bloom. It's also really close to Shibuya, so its easy to hit both in one day. It's home to cat ally and some super killer okonomiyaki places, though the best one requires reservation in advance on certain days and times.
Asakusa is a usual tourist spot with the Sensoji right there. There's a couple other temples there, including one dedicated to those racoon things that fly on their giant balls I forgot the name of. Also some nice little shopping places. I suggest taking the Sumida River cruise if you have time. It's really nice and you can see a good amount of tourist areas without having do to a lot of walking. (And this cool building that looks like a giant beer.) You can also get giant bottles of vodka there for like 500 yen so?
Just outside of Tokyo, about an hour or so train ride away (and relatively cheap to travel to) is Kamakura, which is the home of Daibutsu (the giant buddha) and some really nice little temples and shrines just off the beaten path from that main attraction. A big warning, though, is once you get out of the tourist area there, it's almost 100% japanese unlike tokyo which has a lot of english on the signs and a lot more english speakers. You won't get much help looking for things unless you have a map or something and a lot of pointing.
Also don't plan on getting into any hotsprings or bathhouses if you have visible tattoos.
EDIT: I almost forgot, but if you haven't reserved a hotel I suggest the ANA. It's super nice and also has a bus that goes directly to and from Narita airport for $30. It's in Roppongi which is pretty much at the center of everything so easy travel.
And like everyone else said, don't worry about sticking out. The residents of Tokyo are very forgiving of visitors because they see so many. There were quite a few times when I was trying to politely use Japanese and they whipped out the English to make it easier haha.
EDIT EDIT: I also forgot that most of the vending machines that sell tobacoo and cigarettes require a japanese ID to use in Tokyo now from what I saw in 2010 so.. if you wanna do that you either have to find a nice local or get it from in the store. I'd also like to note I don't think they really check ID since when I went there the first time at 16 I got into night clubs and bars without anyone even blinking an eye.
That corg is in a burrito blanket roll of bread! I want one too now. Watch as the corgi one ends up having the highest rarity from the set and I wind up with a hundred breaded dog gachapon figures. 
