Here is the full list of all 42 Rocket Arena achievements.
Reach Level 100 with any Competitor
Win a Match as Boone in a Social or Ranked Playlist
Win 100 Matches in Social or Ranked Playlists
Win a Match as Topnotch in a Social or Ranked Playlist
Score 50 Rocketballs in Social or Ranked Rocketball
Collect 1000 Coins in Social or Ranked Treasure Hunt
Reach Level 50 with any Competitor
Win a Match as Izell in a Social or Ranked Playlist
Complete a Weekly Challenge
Discover All Artifacts
Reach Level 25 with any Competitor
Megablast 100 Players in Social or Ranked Playlists
Accumulate 1000 Mega Rocket Time in Social or Ranked Mega Rocket
Get 100 KOs as Jayto in Social or Ranked Playlists
Get 100 KOs in Social or Ranked playlists
Get a Great Recovery 50 Times in Social or Ranked Playlists
Earn a KO Assist 100 Times in Social or Ranked Playlists
Reach Level 10 with any Competitor
Complete a Daily Challenge
Equip an Outfit
Equip a MegaBlast or Return Trail
Earn your first Victory in Social or Ranked Playlists
Win a Match as Jayto in a Social or Ranked Playlist
Customize Your Totem
Complete the Tutorial
Reach Level 25 with 5 Competitors
Get 1000 KOs in Social or Ranked Playlists
Win a Match as Amphora in a Social or Ranked Playlist
Win a Match as Blastbeard in a Social or Ranked Playlist
Win a Match as Kayi in a Social or Ranked Playlist
Win a Match as Mysteen in a Social or Ranked Playlist
Win a Match as Plink in a Social or Ranked Playlist
Win a Match as Rev in a Social or Ranked Playlist
Get 100 KOs as Amphora in Social or Ranked Playlists
Get 100 KOs as Blastbeard in Social or Ranked Playlists
Get 100 KOs as Boone in Social or Ranked Playlists
Get 100 KOs as Izell in Social or Ranked Playlists
Get 100 KOs as Kayi in Social or Ranked Playlists
Get 100 KOs as Mysteen in Social or Ranked Playlists
Get 100 KOs as Plink in Social or Ranked Playlists
Get 100 KOs as Rev in Social or Ranked Playlists
Get 100 KOs as Topnotch in Social or Ranked Playlists