Remember that the 53xx series phones use a completely different set of ports than the old 86xx IP phones, if you have never had a MiNET phone external to the system before you may need to reconfigure your router to port forward the appropriate ports, but otherwise it is just default the phone, connect to POE and hold 7, select Yes, IPv4, New, and enter the public IP address of the 5000 system, press VOL DN and confirm to reboot.
Also remember the phone device is different, you can't just "replace" the old one... in the Mitel 5000 you must delete the current extension and create a new one that is the correct type, and since it is remote set the NAT type accordingly.