How can I get Japanese characters to show properly in Firefox on Arch?

It should be enough to install the great noto fonts bundles:

sudo pacman -S noto-fonts-cjk noto-fonts-emoji noto-fonts

The restart firefox and you should be abe to see them. Personally, I also installed the following from AUR:

yaourt -S ttf-freefont ttf-ms-fonts ttf-linux-libertine ttf-dejavu ttf-inconsolata ttf-ubuntu-font-family

I doubt those will help for Japanese, but they do provide a respectable variety of fonts for your system.