Plus or plus hard game modes are unlocked after you complete the game once on normal difficulty.
Plus mode is essentially a brand new game in which all of the enemies and scenarios remain as they were in normal difficulty , you get to keep all of your upgrades you earned in your first playthrough.
Plus hard is the same, although the enemies are more aware, quicker to respond and your health is capped at a single bar. For the purpose of achieving this particular achievement, there is no need to complete every memory in plus hard as doing it on plus will earn the achievement just fine.