Here is the full list of all 40 Princess Kaguya: Legend of the Moon Warrior achievements.
Destroy 100 enemies with kendama stick
Clear stage 1
Defeat HITOTSUME CHOCHIN 100 times
Defeat SHIROUNERI 100 times
Defeat TORIKABUTO 100 times
Defeat KURAYARO 100 times
Defeat USHIONI 20 times
Defeat SHIRONUNO 100 times
Defeat KASHA 100 times
Defeat RABBIT 100 times
Defeat BURARIBI 50 times
Defeat WANIGUCHI DAIOH 10 times
Defeat KYORINRIN 10 times
Defeat KASHA(BOSS) 10 times
Defeat OHMAGADOKI 10 times
Clear stage 2
Clear stage 3
Clear stage 4
Achieve 100 combos
Achieve 150 combos
Achieve 200 combos
Achieve a high score of 100,000
Achieve a high score of 150,000
Achieve a high score of 200,000
Destroy 100 enemies with bow and arrows
Clear all stages on Easy mode
Clear all stages on Normal mode
Clear all stages on Hard mode
Activate KYOGETSU 100 times
Train at the Rabbit Dojo
Defeat 30 opponents at the Rabbit Dojo
Defeat 50 opponents at the Rabbit Dojo
Defeat 70 opponents at the Rabbit Dojo
Defeat 100 opponents at the Rabbit Dojo
Grab 100 healing items
Use special move 100 times with the kendama stick
Use special move 100 times with bow and arrows
Defeat 1,000 spirits
Defeat 3,000 spirits
Defeat 5,000 spirits