The benefit is that you can play him from a smaller domain, and thus likely earlier in the game.
Arcanis has a printed costs of 4, so you could typically only play him using a domain with four resources attached. This would typically be in round three at the earliest.
But as you note, his text says "When you play The Archmage, Arcanis, lower the cost by 1 for each neutral resource attached to the domain drained to pay for him."
In CoC, you don't need to drain a domain that covers the printed cost of a card, only the actual cost. There are plenty of cards that adjust costs of cards out there. A large number of them adjust the cost of "the next card you play" or all cards of a certain type, this one just adjusts the cost of itself, which while rare isn't new (refer to Core set Yog Sothoth).
So, if you had a domain with two Agency resources and one Neutral resource, then you could drain that domain to pay for The Archmage, Arcanis, even though it only has 3 resources.
I concede this is a little weird, since according to the 'Action Window in Detail' section from the FAQ. you are suppose to determine costs before you pay the costs and in the above case you arguably don't know that he costs three until you have already paid that three from that domain. But it is also pretty simple golden rule territory, this card text would be useless if it couldn't actually lower the cost to play the card, so we have to grant it the minimum power needed to do that. The simplest way to grant that power is probably to have the declaration of intent that starts the play action include a declaration of intended domain allowing the correct cost calculation to be made.
But, in practice you more likely just drain a domain, declare the card you are playing and confirm the domain was enough (or rewind with embarrassment if you were wrong).