That's tricky. If the Robe of the Archmagi is a robe enchanted as a +5 suit of armour, then it doesn't stack. If it's an item of clothing which acts as a suit of armour that gives a +5 bonus and happens to be magical, then it stacks. There are two possible schools of thought:
1) The robe's AC bonus is 5 points better than that of a mundane robe. Therefore, it has been enchanted as a +5 suit of armour. As Magic Vestment doesn't stack with magical armour (2 enhancement bonuses), they don't stack.
2) The robe is a Wondrous Item, not a suit of armor. This means that the AC bonus is a property of the wondrous item. As Magic Vestment stacks with these, they stack.
The robe provides an armor bonus, so if it's a legal target, magic vestment improves it.
Magic vestment affects an "armor or shield," so it does not qualify. Magic vestment additionall states it will affect "regular clothing" as though it were a armor with a rating of 0.
Since a robe of the archmagi is not "regular clothing" and can't be construed as +0 bonus armor, it appears the robe of the archmagi is not a valid target for this spell.