There are other options. They could incentivize old people to move to cheaper places. Why should a Japanese person slave away to support a retiree instead of their own family? Why should an immigrant work to support a Japanese retiree instead of their own family?

Temporary and permanent immigration to Japan seems to be increasing in the last few years.

On the "influencer" side, you also are seeing more and more people talk about how cheap houses in beautiful parts of Japan are, how easy / difficult it is to buy and live in one of them, etc. Come to think of it, are these covert advertisements by the Japanese government to get more foreign immigration and capital?

There's only so many countries that will have a surplus of young workers though, and arguably they're ones with cultures/ethnicities least likely to be readily integrated into Japanese society. And such workers are likely to have better options available as other countries with more of a history of accepting immigrants look to remedy their demographic woes. I say that as a Westerner with an ambition to spend some time living/working in Japan - but AFAIK the only way it would be possible currently is to legally marry my (Japanese) partner, so most likely we'll go elsewhere.

Doubt it, they're racist and xenophobic. Much more likely that we see massive entitlement programs for families that are able to inch up the birth rate to a level they can accept.

Where would they get those immigrants from?

It's a high-IQ, high-trust, homogeneous society. They won't get people who can integrate. The most sensible thing would be to promote fertility through incentives, or find other ways to accommodate an aging population.