Buddhism had arrived into Japan in the Heian period as it was taken from China along with a lot of other things that they modelled their society on. I've read works from the period and they sometimes made pilgrimage to various Buddhist temples.

I know that one belief from Japanese Buddhism of that period is that if you have earthly desires or regrets then it's harder for you when you die and your soul will linger. I have no idea about any short of afterlife but the idea of your soul lingering because of the earthly desires is in a lot of literature from the Heian period.

You have to remember that they took Buddhism from the Chinese and then would have changed it to suit themselves. I'm not sure or anything but I wouldn't be surprised if the Buddhism practised in the Heian period isn't the same as Buddhism in Japan now in which case what were those original beliefs that Sai would have practiced?