Geas can be interesting or boring, depending on presentation and methodology. As in the days of yore, when evil DMs like myself twisted the wording of a wish, a player can twist the wording of the geas to try and fulfill their needs as well.
In your case, I'd have the artifact be sentient and telepathic (see intelligent items in DMG). It puts the attuned character under the geas, giving it the vague demand that the PC destroy it's enemy. The player may take some minor negative effect at the end of the first day, since they couldn't possibly do it, then the player can explain (telepathically) that it couldn't do it without more information. You can sprinkle information about the enemy (or at least possible information) among the party's goals, so that they can try to convince the artifact that they are working towards it's goal, while keeping their own.
In your case, I'd have the artifact be sentient and telepathic (see intelligent items in DMG). It puts the attuned character under the geas, giving it the vague demand that the PC destroy it's enemy. The player may take some minor negative effect at the end of the first day, since they couldn't possibly do it, then the player can explain (telepathically) that it couldn't do it without more information. You can sprinkle information about the enemy (or at least possible information) among the party's goals, so that they can try to convince the artifact that they are working towards it's goal, while keeping their own.