The instant or sorcery card you target must have mana value 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.

If the card has in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.

If Dreadhorde Arcanist leaves the battlefield before its triggered ability resolves, use its power as it last existed on the battlefield to check whether the card is still a legal target.

If you cast a card "without paying its mana cost," you can't choose to cast it for any alternative costs. You can, however, pay additional costs. If the card has any mandatory additional costs, you must pay those to cast the card.

If you cast the card, you do so as part of the resolution of Dreadhorde Arcanist's triggered ability. You can't wait to cast it later in the turn. Timing permissions based on a card's type are ignored, and the spell will resolve before blockers are declared.