You choose whether to cast the spell as I Am Duskmourn's ability resolves. If you do, you do so as part of the resolution of the ability. Timing restrictions based on the card's types are ignored.

If you cast a spell for another cost "rather than pay its mana cost," you can't choose to cast it for any alternative costs. You can, however, pay additional costs. If the spell has any mandatory additional costs, such as that of Abhorrent Oculus, those must be paid to cast the spell.

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