If you are unable to cast the targeted card (there are no legal targets for the spell, for example), nothing happens when the ability resolves, and the card remains in its owner's graveyard.

If you cast a card "without paying its mana cost," you can't pay any alternative costs. On the other hand, if the card has additional costs (such as kicker or multikicker), you may pay those.

If you cast the targeted card and it would be put into its owner's graveyard from the stack for any reason (either because it resolves or because it's countered), that card is exiled instead.

If you want to cast the targeted card, you cast it as part of the resolution of Wrexial's triggered ability. Timing restrictions based on the card's type are ignored. Other restrictions are not (such as "Cast [this card] only during your end step").