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

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

You choose whether to cast a spell from among the cards in your hand as Buster Sword's middle ability resolves. You can't wait to cast one later in the turn. Timing restrictions based on the card's types are ignored.