Lacking context, this question will inherently received open-ended answers.
Dragons all share common weaknesses. Ice type, Fairy type, and other Dragon types are all Super Effective against Dragon types. Fairies are also immune to Dragon type attacks. So, in particular, You'd want a Steel or Poison type attack to eliminate them. Other Dragons should be no issue, since your STAB move would be Super Effective against them. The issue is outspeeding them, and ensuring you can KO them faster than they can KO you. Ice types tend to be pretty fragile, so just hit them hard enough and they are out of the way. It's just bonus points if it's Super Effective.
With even attacking stats, 120 in both Atk and SpAtk, you are free to establish a moveset that is either physical or special. But you're probably going to use Judgement, since it's type will match the Plate required to make Arceus Dragon type in the first place, so a Special set would be a natural choice. If that is not to your liking, Arceus has access to Dragon Claw, Dragon Pulse, Outrage, and Draco Meteor. Claw and Pulse are reliable standbys, while Outrage and Meteor have drawbacks in exchange for higher power.
To deal with Fairies, you have a lot of options. If you want Poison, Arceus learns both Sludge Bomb (special) and Poison Jab (physical). If you want to use Steel instead, it has access to Iron Head (physical), Iron Tail (physical), and Flash Cannon (special).
Ice types are easy to deal with as well. The same Steel type moves apply here, but we also have Fire, Rock, and Fighting moves available. Heat Wave (special), Brick Break (physical), Flamethrower (special), Rock Tomb (physical), Overheat (special), Focus Blast (special), Stone Edge (physical), and Rock Slide (physical) are all available.
Arceus is pretty much the definition of flexible. The limiting factors are going to be your intentions with using it.