Here is the full list of all 34 Galagan's Island: Reprymian Rising achievements.
Achieving victory against first boss.
Achieving victory against second boss.
Achieving victory against third boss.
Achieving victory against fourth boss.
Achieving victory against fifth boss.
Anihilate first boss without getting hit.
Anihilate second boss without getting hit.
Anihilate third boss without getting hit.
Anihilate fourth boss without getting hit.
Anihilate fifth boss without getting hit.
Repairing ship 100 times in total.
Upgrading weapon speed 100 times in total.
Upgrading ship speed 100 times in total
Repair the ship when having less than 10% hit points.
Upgrade only weapon speed during one complete run.
Upgrade only ship speed during one complete run.
Activate Super Laser during fight with any Boss.
Take 100 Hits while being protected by a single shield.
Launch Super Laser while being protected by a shield.
Collect 250 Upgrades in total.
Destroy 1000 Enemies.
Destroy 5000 Enemies.
Destroy 15000 Enemies.
Destroy 25000 Enemies.
Destroy 50000 Enemies.
Finish a Game in Co-Op mode.
Borrow a life from another player in Co-Op mode.
Finish 1 level without getting hit.
Finish 5 levels without getting hit.
Have a Double ship for a minute continuously.
Kill 100 Enemies during one run using clone ship attached to the back of your ship.
Finish the game for the first time.
Unlock all other achievements.
Beat the game without your gun ever overheating.