Can I apply for a Chinese visa from Japan with about one month left on my alien registration card?

You can try to apply China visa in Japan even though there is only one month left on your alien registration card. As I know, it is easy for applicants from Canada, USA, UK, Australia to get a China visa in Japan. You'd better go to the Tokyo and apply visa in person, if you can't do it in person, you can ask someone else to do on your behalf.