Here is the full list of all 23 Premium Pool Arena achievements.
Pocket 7 balls in a row
Beat an opponent in a match without them pocketing any of their own target balls
Pocket 100 balls
Pocket two of your own target balls in one shot
Win a match when opponent pockets 8 ball prematurely
Win your first match
Pocket 6 balls in a row
Win 5 matches in a row
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
Play a tournament and claim rewards
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 a game without making a foul
Win 10 games without making fouls
Win 100 games without making fouls