Iceland was able to stay isolated because they are so far away from anyone else. The problem with Japan is that it is so close to China and Korea, and there are a lot of Chinese people in China. So even if they tried to close the borders it's likely that Chinese refugees would end up there and cause an outbreak.
If they did close down all travel routes though it's likely that isolated parts like Okinawa could get by. If they did it's possible that they reverted to the isolationism of medieval Japan. On the other hand they might be exploring the Pacific, thus maybe encountering Americans on Guam and Polynesians in Oceania and on the Easter Islands.
That's always a possibility, though I'm assuming that not only would the JSDF fire upon foreign ships with extreme prejudice, but most refugees would probably be too rushed to effectively prepare, and would perish at sea from a lack of food and water or even rash infection along the way, at least any Chinese. I imagine most people "fleeing" the disease would do as we saw in the prologue and instead head inland.
But even better, try to imagine this scenario:
Monsoon season separating Japan from the rest of the world in the time period where the rash broke down civilization, thus hindering any and all potential infected refugees.
A third Kamikaze, to be specific.
With magic then coming about later, you can imagine a
crazy strong revival of Shintoism in Japan. That would be... interesting.
All famine aside, Japan would then in the first few years after the cataclysm face two options:
-Completely closing off from the rest of the world, not even permitting exploration of neighboring countries.
-Searching outward, possibly out of desperation for overpopulation issues and famine, and then learning the truth of the rash sickness that way.
Human curiosity suggests the latter, though how it would play out is anyone's guess.