To unlock the levels for Arcade mode you must complete each level in Adventure mode on any difficulty setting. In arcade mode the difficulty is set at Expert mode. You will only get three ships and cannot change the difficulty.
Challenges
Successfully complete Mitsurin Jungle to unlock Challenges 1, 2.
Successfully complete Mokuzu Depths to unlock Challenges 3, 4, 5.
Successfully complete Sunahara Desert to unlock Challenges 6, 7.
Successfully complete Hibashira Plains to unlock Challenges- 8, 9, 10.
Successfully complete Sekihi Belt to unlock Challenges 11, 12, 13.
Successfully complete Sekai Outpost to unlock Challenges- 14, 15, 16.
Successfully complete Chuuroh City to unlock Challenges- 17, 18, 19.
Successfully complete Zenshoh Station to unlock Challenges 20, 21 ,22.
Challenge mode bonuses
Chuuroh City concept art: Successfully complete Challenge 18.
Chuuroh City music: Successfully complete Challenge 17.