Right. IF the new cooler system really HAS thermal paste pre-applied, that is ALL you need. Do NOT add more on that cooler OR on the CPU lid. Check very carefully - many that have pre-applied paste also have a thin clear plastic film over that to protect it. That film MUST be removed before installing.
And also to clarify, when you add the new paste (if the cooler does not come with it pre-applied) put the paste on the CPU and mount the cooler on top. It's a lot easier this way, and the most universally recommended.