Glad for you.
They may not know other things to hire a foreign employer too. So, it may be better you know some rules and your rights, to take an initiative when needed.
When your work permit is issued, you go to NIA to get your ARC, and provide the ARC to your employer, so that they can apply for your NHI. I’m not sure, but, labor insurance may need your ARC too.
You don’t need to answer which job you take. I just thought a cram school doesn’t know the procedure is a bit weird, but for a business company not so much.