Here is the full list of all 25 Shmup Arena achievements.
Complete all stages on hard difficulty
Complete the tutorial stage
Complete the first stage on normal difficulty
Complete the second stage on normal difficulty
Complete the third stage on normal difficulty
Complete the fourth stage on normal difficulty
Complete the fifth stage on normal difficulty
Complete the sixth stage on normal difficulty
Complete the seventh stage on normal difficulty
Complete the eighth stage on normal difficulty
Complete the first stage on hard difficulty
Complete the second stage on hard difficulty
Complete the third stage on hard difficulty
Complete the fourth stage on hard difficulty
Complete the fifth stage on hard difficulty
Complete the sixth stage on hard difficulty
Complete the seventh stage on hard difficulty
Complete the eighth stage on hard difficulty
Get a combo greater than 100
Get a combo greater than 250
Get a combo greater than 500
Get a combo greater than 750
Get a combo greater than 1000
Get a combo greater than 1250
Get a combo greater than 1500