Unruly Catapult's triggered ability resolves before the spell that caused it to trigger. It resolves even if that spell is countered.

In a Two-Headed Giant game, Unruly Catapult's activated ability causes the opposing team to lose 2 life.

Players can cast spells and activate abilities (such as Unruly Catapult's own activated ability) after the triggered ability resolves but before the spell that caused it to trigger does.