Not when your course starts, not when your contract begins — when you clear immigration at Incheon or Gimhae. Count forward 90 days from the stamp in your passport and put that date somewhere you will see it.
Appointment slots at busy offices (Seoul, Suwon, Ansan) can be weeks out, so book as soon as you have an address. Booking early costs nothing; discovering in week eleven that the next slot is in week fourteen is a fine you did not need to pay.