They will need to renounce their other citizenship immediately. JPN will require the renunciation letter before issuing the MyKad.
Once you have received the citizenship certificate, you can proceed to JPN to apply for the IC and wait for updates. The process may include an interview and other steps. You can expect a call from JPN notifying you that the IC has been approved, at which point you will be required to renounce the other citizenship.
To do this, visit the embassy of the relevant country to begin the renunciation process. Each country has different timelines, fees, and procedures for renunciation, so it is advisable to check with the relevant embassy for specific details. Once you receive a renunciation confirmation letter from the embassy, submit the letter to JPN for the MyKad to be issued.