Here is the full list of all 60 502's Arcade achievements.
Pet the 4 different colors of cats
Make a 5X combo
Make a 9X combo
Make a 15X combo
Destroy 50 pots
Hit spirits with projectiles 100 times
Equip all projectiles during a game
Seal 100 spirits
Seal 300 spirits
Seal 500 spirits
Defeat the Oni
Defeat the Nukekubi
Defeat the Waira
Defeat the Tengu
Complete a loop finishing all levels with an S rank
Make a 2 potions chain
Make a 3 potions chain
Make a 5 potions chain
Play 100 times
Stay on the "danger zone" for more than 30 seconds
Make 100 potions
Hard worker
Make 500 potions
Prepare 100 recipes
Prepare 300 recipes
Prepare 500 recipes
Prepare 1000 recipes
Reach Level 5
Reach Level 10
Reach Level 15
Defeat 200 enemies
Defeat 500 enemies
Defeat 150 enemies with the flamethrower
Defeat 65 enemies with the grenade launcher
Defeat 80 enemies with the rocket launcher
Defeat the first area's boss
Defeat the second area's boss
Defeat the third area's boss
Complete the first area without losing health
Complete the second area without losing health
Complete a loop without losing any health
Defeat at least one enemy of each kind
Use all weapons in the game
Complete a loop with every character
Complete a second loop of the game
Defeat 700 enemies
Throw 200 enemies into the water
Defeat 200 enemies while they're holding a chest
Throw 125 objects or enemies into other enemies
Defeat 140 enemies using the lightning attack
Defeat the first boss
Defeat the second boss
Defeat the third boss
Buy all upgrades at least once
Keep all your treasure chests until round 10
Get 3 helper parrots during the same run
Defeat 70 enemies using the portable cannon
Defeat 40 special shiny enemies
Recover 3 chests during the same run
Complete a second loop