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

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.

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