This tutorial will explain how you can easily capture Specter.
When dealing with the waves of enemies, use your spin attacks to quickly get rid of them. For the bunnies, wait until they finish their stomp attacks before attacking. For the slime and spike enemies, attack them quickly before they get a chance to hurt you.
When you grab the rocket, stay in the same position while facing him. This is where he will fall when you shoot him.
Once he fell down to the ground, spam R2 to smash your net at him. You should be able to capture him before he gets the chance to run away.