Here is the full list of all 23 Premium Pool Arena achievements.
Win a game without making a foul
Pocket 100 balls
Pocket 7 balls in a row
Pocket 6 balls in a row
Win your first match
Win a match when opponent pockets 8 ball prematurely
Play a tournament and claim rewards
Pocket two of your own target balls in one shot
Win 5 matches in a row
Beat an opponent in a match without them pocketing any of their own target balls
Pocket 1000 balls
Play 100 matches (until the game over screen)
Play 1000 matches (until the game over screen)
Win 10 matches in a row
Win a game by pocketing all balls in a row in 8 Ball
Pocket 10000 balls
Own 10 cues
Achieve level 10 in a tournament
Win a tournament without losing a match
Play 100 matches in the tournament mode
Play 500 matches in the tournament mode
Win 10 games without making fouls
Win 100 games without making fouls