As versatile as Arceus is, every individual set is entirely counterable. Giratina and Lugia can switch in on an Extreme Killer set very easily with their ridiculous Defense and HP and burn or Whirlwind it away, respectively; additionally, intelligent switching lets them use their Pressure ability to disintegrate its meager 8 ExtremeSpeed PP. Skarmory works similarly to Lugia, although the Overheat version squashes it. However, Skarmory is also capable of dealing with the Ghost Swords Dancer, a distinction shared by the insanely defensive Normal Arceus, which can Will-O-Wisp or inflict serious damage on both species of Arceus. A very defensive Choice Band Groudon can also counter the Swords Dance Ghost Arceus, barring a critical hit from Shadow Claw. While not a counter in adherence to its strictest definition, a Kyogre with Choice Scarf can switch in after it has knocked something out, and then proceed to land a swift OHKO with Water Spout, assuming that it did not suffer from damage beforehand.
All counters to the Swords Dance Ghost Arceus except the Normal Arceus must caution against walking directly into a Calm Mind version, though. Speaking of the Calm Mind versions with an auxiliary attack, Blissey can use Toxic to wear them down if they have Recover; if they have Substitute, that means they lack any recovery whatsoever, which a Calm Minding Blissey can counter by breaking the Substitutes down slowly. If the Calm Mind sweeper is Ghost, then a Normal Arceus will be able to switch in on its relatively weak Ice Beams and use Toxic or Punishment, while a Bug-type sweeper is devastated by Heatran and Ho-oh, which sport excellent resistances to its attacks and do considerable chunks of damage with Fire-type attacks that are likely boosted by sunny weather.
Latias and Latios with Refresh can quickly decimate Fighting Arceus as well as Grass Arceus. Rest and Sleep Talk Giratina is in the same boat as Latias. With its base 120 Special Defense, Giratina does not fear much from either of those, barring a critical hit or freeze from Ice Beam. With Pressure on its side, Giratina can quickly dispose of Ice Beam's PPs, leaving Arceus with absolutely no weapons against Giratina. However, Giratina must be cautious walking into versions with Calm Mind, as they will turn the tables quite easily. Although it fears repeatedly switching into STABed Judgments, Blissey can come into either of Grass or Fighting Arceus, land its own Toxic, Softboiled and switch out before she takes too much from Toxic if Arceus inflicted it on Blissey upon switching in. Dialga and Lugia can handle the Grass variant fairly nicely; the former can Bulk Up and dispatch Arceus with a powered up Dragon Claw, whereas the latter can Toxic and Whirlwind away Grass Arceus out of the battlefield. Metagross finds itself not pressured to switch into Grass Arceus and 2HKO with Choice Band Ice Punch or Meteor Mash. Heracross does not fear any of its attacks, and swiftly disposes of Arceus with Megahorn. Heatran, with its 4x resistance to both Ice and Grass, can quickly demolish Arceus with powerful STABed Choice Specs Flamethrower.
The mono-attacking sets are a bit complicated to counter as it is difficult to inflict any type of lasting damage due to the combination of Substitute and Recover or Rest and Sleep Talk. The Bug-types are the easiest to counter; Giratina with Roar can remove any stat boosts as well as a possible Substitute, while Ho-oh and Heatran sport quadruple resistances to Bug and powerful Fire-type attacks to retaliate with. The Ice-type special sweeper is relatively easy to deal with, as well; the Fire-types still work, and Kyogre with Calm Mind rends it apart due to resistance to Ice and an amazing base 140 Special Defense. Metagross can switch in and use a Choice Banded Meteor Mash to easily dispose of Arceus. Lastly, the Dragon-type is countered by Metagross.
The Choice Band set is relatively simple to counter. Giratina can switch into any attack, even a Dragon Claw, and then one may work on switching in Pokémon that resist the chosen attack; to ease matters, Giratina's Pressure ability ruins ExtremeSpeed and Punishment. Lugia and its massive Defense, Reflect, Pressure and Roost does the same too. The Arceus that are designed to counter other Pokémon are usually beaten by a solid attacker that is not weak to one of their attacks. Any non-Poison or Steel-type must beware of Toxic, however.