What should we do if Japan aged it's self out to the point where they cannot regenerate and recover from the inevitable, and they went away?

Your original error - "it's self out" - corrected by the second error - "aged themselves out" is very telling.  Very quasi-intelligent question.  It will never happen.

There's no such thing as "aging yourself out".

The habitable parts of Japan have a very high population-density. Japanese per-capita consumption is 10 times their fair share of what nature (worldwide) could grow back. The population shrinking because fewer babies are born than people die would be a good thing.  Even with Japanese fertility going a little below the current 1.4 children per woman, there isn't much we would have to do. There are thousands of ethnicities that are way closer to extinction than Japanese

Where to even begin with this silliness? "Itself." Not "it's self." Its is possessive. It's is a contraction of "it is." Please learn basic English. Even with a negative birth rate it would still take many, many, many years for a country with a population of 130,000,000 to shrink down to a population of zero. And obviously people would begin to take drastic action if the numbers ever got dangerously low. The Japanese would simply have to face the reality that they have been avoiding for decades: that immigration is the only thing that is going to preserve the standard of living in the developed world. Imagine if the Japanese government were to ease immigration restrictions for people from places like Cambodia, Indonesia, Malaysia, Taiwan, Thailand, Vietnam, and others. Imagine if people who are relatively well educated, who have a bit of savings, and who are at the top of their game in their own country were admitted to Japan, granted residency with the possibility of citizenship and given the opportunity to contribute to Japanese society. Sure, it would take them twenty years to become fully Japanese, and they'd be old and grey by then, but their kids wouldn't. Their kids would be Japanese in five or ten years. And the kids of those kids would be Japanese at birth. That's how we solve our population problems in the West, and that's how the Japanese are eventually gong to fix theirs. Please try to use your noggin for something besides a decoration. 

Time to have more sex with Japanese