How common is it for a Japanese woman to marry a White American Airman?

Question:

I happen to think they are gorgeous and really want to know if I would ever have a shot.

Answer:

It is not common by any means. I would say 99% of Japanese women have no desire to get married with US military personnel, because (speaking in general),

1) they don't like US military in Japan for many historical reasons;
2) they don't want to get drug around the world every time you have new assignments in strange countries (this makes a big difference with many military wives in the US who can stay home in North Dakota (or wherever) to wait for your home-coming dead or alive);
3) there would be no financial security once you get out of the service;
4) majority of NCOs don't have a college education which is the minimum requirement for a decent job in Japan, which makes them feel insecure about their future (with you in Japan after your retirement).

As you know well, women are more "realistic" than men wherever you go. Men are dreamers, but women are not, which is more true with Japanese women than Americans. You should keep that in mind.

Edit:
You may think what I mean by the reason #2 above is unclear.
Let me explain: As self-declared "police of the world," the US stations its military in more than a few dozen countries around the glove. You never know when you get a new order to station in an ex-USSR country in central Asia that you've never heard of. For how long? You'd never know, either. It's up to 'the uncertainty' of the international situation.

By then, you may have some kids with your Japanese wife (if you were criminally lucky) who would be left in a small apartment, if not a chicken shuck (thanks to USFJ housing allowance which is supported by 'compassionate' Japanese tax payers) outside the gate of Misawa Air Base close to the nuclear power plant in Rokkasho-mura in Aomori prefecture which is prone to another big earthquake and tsunami.

If you were compassionate and responsible enough to take care of your lovely wife and kids for the rest of your life, you may need to consider the possibility to bring them back to your shabby-looking small village in North Dakota. Most probably, she wouldn't accept it. Then, you need to consider the possibility to get out of the service. Well, if so, we need to go back to the reason #3 and #4 again. Clear enough?

Edit 2 (in response to your "Additional Details."
:
I apologize if I hurt your. But I sopke "in general" as I clearly said above before I wrote what you call this "nonsense." I am totally aware that there are many exceptional cases; otherwise, I didn't have to write "speaking in general," did I? Whatever you say, I am certain that this is our "general" image of White (brown, black or yellow) American Airman," speaking in general.

BTW, mechanic is highly respected profession in Japan, too. But, there are millions of excellent mechanics in aviation and automobile industries in Japan. I am sure you can survive in the US (or somewhere else) but not in Japan. "Your wife" knows that.