If it is an Arizona non resident permit then you will not be able to carry, openly or otherwise, in a vehicle in Michigan. Furthermore you may not carry concealed under the non resident permit because to carry concealed in Michigan you have to be licensed by this state or your state of residency to carry a concealed firearm.
To transport a firearm in Michigan without a CPL you must comply with MCL 750.231a,
MCL 750.227 concerns vehicle carry and the concealed carry of a firearm in this state.
750.231a Exceptions to MCL 750.227(2); definitions.
(1) Subsection (2) of section 227 does not apply to any of the following:
(a) To a person holding a valid license to carry a pistol concealed upon his or her person issued by his or her state of residence except where the pistol is carried in nonconformance with a restriction appearing on the license.
(b) To the regular and ordinary transportation of pistols as merchandise by an authorized agent of a person licensed to manufacture firearms.
(c) To a person carrying an antique firearm as defined in subsection (2), completely unloaded in a closed case or container designed for the storage of firearms in the trunk of a vehicle.
(d) To a person while transporting a pistol for a lawful purpose that is licensed by the owner or occupant of the motor vehicle in compliance with section 2 of 1927 PA 372, MCL 28.422, and the pistol is unloaded in a closed case designed for the storage of firearms in the trunk of the vehicle.
(e) To a person while transporting a pistol for a lawful purpose that is licensed by the owner or occupant of the motor vehicle in compliance with section 2 of 1927 PA 372, MCL 28.422, and the pistol is unloaded in a closed case designed for the storage of firearms in a vehicle that does not have a trunk and is not readily accessible to the occupants of the vehicle.
(2) As used in this section:
(a) "Antique firearm" means either of the following:
(i) A firearm not designed or redesigned for using rimfire or conventional center fire ignition with fixed ammunition and manufactured in or before 1898, including a matchlock, flintlock, percussion cap, or similar type of ignition system or replica of such a firearm, whether actually manufactured before or after 1898.
(ii) A firearm using fixed ammunition manufactured in or before 1898, for which ammunition is no longer manufactured in the United States and is not readily available in the ordinary channels of commercial trade.
(b) "Lawful purpose" includes the following:
(i) While en route to or from a hunting or target shooting area.
(ii) While transporting a pistol en route to or from his or her home or place of business and place of repair.
(iii) While moving goods from 1 place of abode or business to another place of abode or business.
(iv) While transporting a licensed pistol en route to or from a law enforcement agency or for the purpose of having a law enforcement official take possession of the weapon.
(v) While en route to or from his or her abode or place of business and a gun show or places of purchase or sale.
(vi) While en route to or from his or her abode to a public shooting facility or public land where discharge of firearms is permitted by law, rule, regulation, or local ordinance.
(vii) While en route to or from his or her abode to a private property location where the pistol is to be used as is permitted by law, rule, regulation, or local ordinance.
You do not have to be the registered owner of the vehicle, you just need to comply with all state law.
Also for the purpose of transportation in a case in the trunk of a vehicle, no it does not have to be locked, and no the magazine does not need to be separated, and no the ammunition does not need to be separated either.
Ex: You unload the firearm OUTSIDE (this being important) of your vehicle then placing the unloaded firearm in the case along with the now loaded magazine, closing the case and placing the case inside the trunk of the vehicle.