Here is the full list of all 16 Contra Anniversary Collection achievements.
Complete all games
Complete Super Contra
Play each game
Complete Contra (Arcade)
Complete Operation C
Complete Super Probotector: Alien Rebels
Clear Route A in Contra: Hard Corps
Clear Route C in Contra: Hard Corps
Clear Route D in Contra: Hard Corps
Complete Probotector
Complete Contra (JP)
Clear Route B in Contra: Hard Corps
Clear Route E in Contra: Hard Corps
Complete Contra III: The Alien Wars
Complete Super C
Complete Contra (NA)