I live in Japan and work for a Japanese company. I see some dubious advice from people who don't even live in Japan on this thread... tread carefully. Anyhow, this is what I would recommend.
If you are invited to nomikai, I do not recommend that you decline outright. You should participate because it will help with relationship building. You absolutely do NOT need to drink alcohol. However you will need to drink something, otherwise you won't be able to participate in the 'kanpai' toast and the drink pouring activities thereafter with your colleagues. In the past oolong-tea was the go-to beverage for people who did not drink. This is fine, but it makes you stand out because the color and appearance are very different beer. I would recommend non-alcoholic beer instead, which should be standard fare at any izakaya (Japanese pub). The flavor of Japanese non-alcoholic beer has drastically improved, if that makes any difference to you, and the appearance is like real beer.
Regarding pub-hopping, I highly, highly recommend that you do not leave early. At the very least, you should stay until the end of the first leg of the event. This is because it is very likely that a manager or somebody of importance will say a few words in closing and possibly do a group 'clap' or something similar that involves everyone's participation. After this, feel free to go back to your hotel. You do NOT need to go to the next pub with everybody else, i.e. nijikai.
Regarding the AA perspective: I can almost guarantee that you will not be going to a 'bar'. You will be going to an 'izakaya'. This is basically a restaurant that serves a wide variety of sharable Japanese (and possibly Western) dishes with alcohol also available. Food and drink are intertwined in Japanese culture, and you should think of them as a set. People do not normally go out to just drink alcohol (unlike the West).
AA is relatively unknown in Japan. If you try to explain it you probably won't be understood. Just say that you don't drink alcohol--that should be sufficient. The number of young people that do not drink alcohol has been increasing in recent years. It is a very real phenomenon called "アルコール離れ" (arukorubanare). The choice to not drink alcohol, whether for health reasons, religious reasons, or philosophical reasons is not a strange thing anymore.
It bears emphasizing: the alcohol is not the important thing, it is the social bonds that are formed by going out eating and drinking together. What you choose to drink does not need to be beer or alcohol. In present day Japan there are many, many alcohol-free options that you can partake in. Maybe you should tell the event organizer in advance that you do not drink alcohol and would like to go to an izakaya with lots of non-alcohol options. This would sound like a perfectly reasonable request to me.
One last thing... even if you do not drink, there will be alcohol and some of the people around you will be drunk. This is just part of the Japanese 'nomikai' experience. I doubt you will see any aggressive drunks though (unless they are foreigners). If this sort of situation--being around alcohol and people who are drunk--might be psychologically stressing for you, then maybe you'll have to take a different approach from the one I described. Your psychological well being should take priority over socializing and social bonding.
I hope that this helps.