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.