They have a handgun ban there. It would not survive a court challenge, but it hasn't been challenged. The only way to carry a firearm there would be under LEOSA.
I know a while ago, Gray Peterson (from this site) emailed the public defender in the CNMI showing him Heller and McDonald. He never got a reply.
American Samoa has the same problem, they have a complete firearm ban except for .410 shotguns and 22 caliber rifles, although LEOSA applies there as well.
All territories and possessions are under the consitution. There was definitive case law on that from the 1910's and the 1970's. I don't remember the case law, it's been a while since I've looked at that.
I do remember that a case out of Puerto Rico stated that fundamental rights (which McDonald clearly stated) apply in all of the United States. It doesn't matter if the territory is a Commonwealth, an unincorporated territory, or a possession. The Bill of Rights applies in them all.
In 1990, Guam passed a very restrictive abortion law, it was struck down a month later under Roe v Wade case law.
To be honest, the territories are not a concern to most people. The licensing officials in the Mariana's asked the NRA for help in writing their Weapons ID Card application a few years ago and they got no response!!!
I do know the CNMI are extremely corrupt and probably won't care about case law, they would most likely have to be sued. There is a website called Saipansucks. It was famous years ago, it is believed that the website was created by a former Assistant Attorney General for the Commonwealth. He has many examples of how corrupt the place is so I doubt they will review McDonald and rewrite their weapons laws.... same for American Samoa.