When does Dreadhorde Arcanist check it's power when resolving the trigger in this situation?
You would only be able to cast a spell with a converted mana cost of 1.

This is confirmed by the rulings on Dreadhorde Arcanist:


  The instant or sorcery card you target must have converted mana cost less than or equal to Dreadhorde Arcanist’s power immediately after Dreadhorde Arcanist has attacked. Any other abilities that trigger when creatures you control attack won’t have resolved yet.


This is because you must choose targets when you put the ability on the stack, before it resolves, and at the time you are choosing targets, Dreadhorde Arcanist still has a power of 1. You will be placing both triggered abilities on the stack before either one of them resolves.


  603.3. Once an ability has triggered, its controller puts it on the stack as an object that's not a card the next time a player would receive priority.


Note that targets are checked for legality again when a spell or ability resolves, so Dreadhorde Arcanist must still have a high enough power to cast the chosen spell when its ability resolves. If the Arcanist's power were lowered after its ability was put on the stack, this could cause its ability to fizzle (not resolve).


  608.2b If the spell or ability specifies targets, it checks whether the targets are still legal. [...] If all its targets, for every instance of the word "target," are now illegal, the spell or ability doesn't resolve. It's removed from the stack and, if it's a spell, put into its owner's graveyard.