plan, plan, plan
This sounds like really general advice. However, in Tokyo planning is more important than your average vacation. A typical tourist will not be able to read or speak Japanese. This makes getting lost very easy and makes simple tasks such as ordering from a menu a big challenge.
Tokyo is a big place and there are generally no street names. The Japanese system of addresses is very complex. Even the locals will use a map to seek out a recommended restaurant.
Sure, it is fun to wander aimlessly when you are on vacation. However, if you don't plan you will end up eating at McDonalds because they are easy to find. You won't see all the coolest stuff and you will miss chances to interact with the locals.
take the train
When in Japan, do as the Japanese. I had an American friend visit who was obsessed with trying to drive everywhere. It's easier to get everywhere in Japan by train.
pack light and portable
Japanese trains are not great for big luggage.
I feel really sorry for tourists with huge bags in rush hour traffic. There are delivery services at the airport that will send your bags to your hotel with 24 hours. It is great to have the freedom of no luggage when your on the train. You don't want to have your first experience in Tokyo being a bad one.
You gotta try it when your in Tokyo. They're everywhere.
Try to remember the characters カラオケ. Find a place that looks new and modern and they will be more likely to have English songs. There is often an all you can drink course with unlimited alcohol. You will likely have trouble working the remote control but the staff will kindly help you. If you have Japanese friends invite them along because Karaoke can present various language challenges.
meet the locals
If you have a local friend your trip will go much smoother. Be nice to them and they will help you unlock great Tokyo experiences. If you don't have local friends be open and friendly and you may make some. Japanese people are very curious about what foreigners think about Japan. Don't complain about Japan and they will like you more.
Tokyo is a drinking town. It is the national recreation of the Japanese. If you don't drink then at least go out to some of the drinking spots to see this side of life.
go in small places
Tokyo has lots of tiny bars, restaurants, cafes and shops. These are the best places so don't be afraid to go in. There are lots of tiny bars that have just 3 to 8 seats. These are the places where everyone in the bar talks and you can get to know the regulars.
Check the web for events in Tokyo before you go. There are hundreds of matsuri (festivals) held in Tokyo each year. Odds are good there will be some events when you are visiting. Matsuri are fun.
get out of Tokyo
Don't come to Japan for two weeks and see nothing but Tokyo bars. You will regret it. If you have time the best place to see is Kyoto. Kyoto takes at least two days.
If you are pressed for time go on a day trip to Kamakura, Hakone or Nikko.
Tokyo has to be the best place in the world for shopping. So check it out.
Akihabara and Shinjuku are good for electronics.
Ginza and Omotesando are good for luxury brands. Even if you don't have a big budget these are great places to walk around and have lunch.
Asakusa is good for Japanese souvenirs (check out Sensoji temple).
Shibuya or Shinjuku are good for clothing and have many Japanese Department Stores.
Shimokitazawa and Harajuku have plenty of weird little shops. If you're a hipster you'll never want to leave.
Check out 100 yen shops. They have cool stuff and most have souvenirs. There are some good ones in Harajuku.
Many expensive Japanese restaurants have great lunch specials for office workers. Places that cost hundreds of dollars for dinner might cost ten dollars for lunch. Avoid the crowds by going at 11:45 or 12:45. They might be closed by about 2pm.
Social norms are very different in Japan. Something that might be rude at home could be perfectly polite in Japan and vica versa. People might push you to get off the train etc.. Don't let it bother you ... you're on vacation. Relax!
Yoyogi Park on Sunday
A lot of people migrate to Yoyogi Park on Sunday to drink beer, listen to music and enjoy the day. You will see weird stuff like girls belly dancing to bagpipes etc... Yoyogi Park is at Harajuku station and there are lots of interesting stops and restaurants in Harajuku.
Japanese convenience stores are great
They have good fresh food and every service you can imagine. Locals use them at least twice a day.
use the correct exit
If you go to a station it is very important to get off at the right exit. One exit may lead to an interesting shopping street while another leads to a endless maze of boring residential housing. If you are going somewhere specific you need to know exactly how to get there. For example, Shinjuku station is the busiest station in the world and has over 200 exits. If you are looking for the shopping district and you take the wrong exit you will end up hopelessly lost.
get a phone
Mobile phones can be rented for a reasonable price at Narita Airport. If you have to meet up with friends it will be important to have one.
I have seen that a lot of foreign phones that can roam to virtually any country in the world do not work in Japan. This situation may be improving recently.
get a subway pass
The first thing you need in Tokyo is a subway pass. There are two types: Pasmo and Suica. Both of them work on all trains in Tokyo and most buses. You can also use them to buy stuff from some vending machines and convenience stores.
Cards can be bought from machines at any train station. The cards have a 500 yen deposit.
plastic models are your friend
Many restaurants have plastic models of their menu outside the restaurant. Check these before you go in and decide what you want. The menu may be completely Kanji without pictures. In the worst case, you can always ask the staff to come outside with you and you can point at what you want.
Japanese culture has no concept of tipping. If you tip they might run after you thinking you forgot your change. In my opinion, Japanese service is the best in the world. However, tipping will just lead to embarrassment and awkwardness. Show respect by following the local customs.
Many foreigners who live in Japan come to love this system. Tipping is so political and uncomfortable. I wish restaurants in the US would charge more for the food and pay their employees decent salaries.
Staff at Japanese restaurants will not come by your table unless you say sumimasen (excuse me). You can make all the eye contact you want and they will not come to your table unless you use this magic word. If it is a loud and crowded spot it is perfectly polite to yell sumimasen!!
This is really obvious advice but it is amazing how many people don't do it. When you are speaking to people in English speak very slowly, gesture a lot and use a simple vocabulary.
good travel karma
When you travel you are representing your country. Enjoy, but don't be disrespectful or obnoxious. Japanese people tend to be really polite when they travel, so return the favor.