Here is the full list of all 29 Samurai Aces achievements.
Stage 4 clear
Stage 3 clear
Stage 2 clear
Stage 1 clear
Upgrade the shot until maximum
Watch ending with Jane
Watch ending with Tengai
Watch ending with Kenno Maru
Watch ending with Koyori
Watch ending with Aine
Watch ending with Gennai
Stage 5 clear
Stage 6 clear
Watch the ending without continuing in EASY Mode
Watch the ending without continuing in NORMAL Mode
Watch the ending without continuing in HARD Mode
Collect maximum quantity of bomb (default setting)
Increased life
Obtain more than 50 bonus items
Scored 1,000,000 with Jane
Scored 1,000,000 with Tengai
Scored 1,000,000 with Koyori
Scored 1,000,000 with Kenno Maru
Scored 1,000,000 with Aine
Scored 1,000,000 with Gennai
Unlock all achievement
Stage 7 clear
Watch the ending without continuing in VERY HARD Mode
Clear the game without destroying any enemies.