If I get a 12 month Visa now, will it still be valid after my ARC runs out?

I'm afraid you won't obtain a China visa in south korea at all since your South Korean Alien Registration Card is valid for less than 6 months.
But if your South Korean Alien Registration Card is valid for more than 6 months and you successfully obtain a China visa for 12 months, you can enter China with no problem even if your South Korean Alien Registration Card is valid for less than 6 months by then.
But the thing is it is difficult to obtain a China visa valid for 12 months in a third country.

I am desperate! How difficult is it? I'm doing my research. What makes it so hard. What do you suggest I do?

Thanks!

You know what? I'm confused about the time of validity left on your South Korean Alien Registration Card, so I tell two possibilities in my last reply. Let me state it more clearly.
1. If your South Korean Alien Registration Card is valid for less than 6 months at present time, you cannot obtain any China visa in South Korea. In this case, maybe you can travel to another country to obtain a China tourist visa, like Japan. But there is a great chance it will be a single entry visa. The visa with 12 months' validity is usually multiple-entry one, which is very diffucult obtaining outside your home country.
2. If your South Korean Alien Registration Card is valid for more than 6 months presently, it's possible for you to get a multiple-entry China visa there. It doesn't matter if the Alien Registration Card will be valid for less than 6 months when you enter China.