Here is the full list of all 24 Star-Pit Starship achievements.
Earn 100,000 Graps Lifetime
Destroy 500 Enemies
Destroy 1,000 Enemies
Build a Ship Containing 1-4 Star-Chips
Build a Ship Containing 5-10 Star-Chips
Build a Ship Containing 11-15 Star-Chips
Build a Ship Containing 16-20 Star-Chips
Build a Ship Containing 21-25 Star-Chips
Build a Ship Containing 26-41 Star-Chips
Build a Ship Containing 42 or More Star-Chips
Destroy a Dark Nag Class Enemy Ship
Destroy a Pack Scrapper Class Enemy Ship
Destroy a Fortune Hazard Class Enemy Ship
Destroy a Noslim's Revenge Class Enemy Ship
Destroy a Brogand's Might Class Enemy Ship
Destroy a Goliarth's Hammer Class Enemy Ship
Destroy a Crator's Throne Class Enemy Ship
Build a Ship Containing at Least 1 of Each Star-Chip
Earn 1,000 Points in a Single Round
Earn 5,000 Points in a Single Round
Earn 15,000 Points in a Single Round
Earn 30,000 Points in a Single Round
End the Round Like a True Champion--Using Self Destruct
Earn 50,000 Points in a Single Round