I have been reading about the item and got curious: if I am neutral, for example, and I am attuned to a Robe of the Archmagi, will it turn from grey to black if I turn evil?
No, the item does not change
The item will remain either white, grey or black. If the character currently attuned to it changes alignment such that it no longer meets the attunement prerequisites of the item, they are no longer attuned nor can they become attuned again until they meet those requirements again (i.e. change back to the correct alignment).
From the DMG, pg 138 under Attunement:
A creature's attunement to an item ends if the creature no longer satisfies the prerequisite for attunement...