Professor Hojo's last ability resolves before the activated ability that caused it to trigger. It resolves even if that activated ability is countered or otherwise leaves the stack.

Professor Hojo's first ability reduces only the amount of generic mana in the first activated ability you activate during your turn that targets a creature. For example, it will reduce an activation cost of {2} to {0}, but it will have no effect on an activation cost of {G}.

Activated abilities contain a colon. They're generally written "[Cost]: [Effect]." Some keyword abilities (such as equip) are activated abilities and will have a colon in their reminder text.