Another document that you might need to get is a Certificate of Participation. Essentially, this is a bilingual document that certifies that you worked for your contracting organization on the JET Programme for your contract period. If you need one, contact your Contracting Organization and ask them for a Zaishoku Shoumeisho (在職証明書)
If you want to know what it looks like, you can download a sample from Clair’s website at Certificate of Participation.