Mighty Goose (JP)

New game plus is first unlocked by completing the 9th and final mission of the campaign called "Kolos 1" (although this is called Level 5 for it's achievement). A portal opens up to a mirror universe of all the same missions but on a harder difficulty.

You do not have to beat all 9 missions again on new game plus for this achievement, only the levels that are considered 'main missions', which are the ones that give you the achievements 'complete level 1, 2' etc.

These are:

1. Void Prison+
2. Planet Zandbak+
3. Bugga'veo+
4. Aether Realm+
5. KOLOS 1+
(simply skip any mission that is not on this list for now)

To make this a breeze, in the armory equip Vark CRT and Protein Bar on the first page, giving you a flying ai supporting you and longer mighty meter usages.
For your secondary weapon use Void King Glove - this is op, use it non-stop whenever it's available every 5 seconds.

Once you have beaten these 5 levels, return to the portal at the start of the mirror universe map, and you'll unlock god mode. This makes you invincible and gives you infinite ammo, secondary usages and mighty meter. This makes getting S rank a breeze as you can just walk right whilst holding X to fire as well as spamming Y non-stop the whole level using the Void King Glove. This is why I highly recommend only doing the 5 NG+ levels listed above first before the ones in between because God mode will make those a cake walk.