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, such as sacrificing a creature, those must be paid to cast the card.

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

If you wish to cast the exiled card without paying its mana cost, you must cast it while Solstice Revelations is resolving. You can't cast it that way later in the turn. A spell cast this way may be cast at a time you normally wouldn't be able to cast a spell of that type, but other restrictions (such as "Cast this spell only during combat") are enforced.