Hi Smythe, if food is your thing than you will LOVE Tokyo!
Hit the Tsukiji Market for Sushi and to experience the crazy outer market. If I were you I wouldn't leave it to the last day, especially if you want to eat at one of the famous sushi joints like Sushi Dai. The lines at the really good places can be long, up to 3 hours if you aren't one of the first people there at the crack of dawn. If it doesn't work into your schedule than try one of the other fab sushi places like Standing Sushi, which isn't expensive and they have a few locations in different neighbourhoods. Also the sushi was almost as good as Sushi Dai in the market. For lunch go to Ramen street (located in the complex attached to Tokyo Station). All the ramen restaurants located in the stretch of hallway (called Ramen street) are well know, popular restaurants so you can't go wrong. If there isn't a line up at Rokurinsha go there, it's probably the most popular. There is also tempura, tonkatsu, soba, udon and kabayaki... hope you're hungry

Personally I think some of the can't miss neighbourhoods in Tokyo are Harajuku, Shinjuku and Shibuya. Like cockpitvisit said in the post above, Meiji Shrine is close to Harajuku so you could do the Shrine and than walk Harajuku in a few hours. Shinjuku has really great department stores with crazy food halls in the basements where you can get amazing prepared foods (this could also work for a dinner or lunch) You can get bento boxes, tempura, and the pastries and sweets rival ones in Paris. Isetan Department store in Shinjuku is a great one. Shibuya is great to people watch and wonder around taking in Tokyo.
The Imperial Palace is just ok... I would skip it at this time of year because the gardens won't be very impressive. (sorry if that offends anyone, I just don't think it's fantastic enough during the winter to dedicate a morning if you are ONLY going to be there one day- just my personal opinion)
If I were you I would save the morning of your last day to hit a museum (Edo or Tokyo National Museum are both pretty good... or the Ramen museum if you are a TOTAL foodie lol) That way you won't have to worry about navigating too many neighbourhoods and subways, I think it might be less stressful since you have to get to the airport in the afternoon.
If you have an iphone and internet at your hotel you can plot a destination and get directions (plus you can drop a pin for one more place) I disables my data and was still able to view the map and use the compass and GPS in the phone to navigate around. I looked at getting a GPS but I couldn't find maps at a reasonable price and the ones I did find seemed a little inaccurate. Google maps worked pretty well for me on my phone though. If you get lost in any neighbourhood though you can ask just about anyone to help you get back to the main subway you arrived at, usually someone will be able to point you in the right direction.
I hope that helped a little, Tokyo is amazing you will have fun no matter what you do.