This specifically mentions "Battle Arenas" which
doesn't include the Dark Arts Battle Arenas as part of the achievement. Which is great news, because there's only two Battle Arenas located on the map!
#1 The
North Ford Bog Battle Arena - head North East from the
"East North Ford Bog" Floo Flame.
#2 The
Feldcroft Region Battle Arena - head South West from the
"Irondale" Floo Flame.
To open the Battle Arena door you'll need to destroy all 20 white Urns that litter the area. The easiest method to spot these is using Revelio, which will highlight them in blue. Once the door is open, head inside to complete 5 waves of enemies that get progressively harder. If you are struggling with the waves go into the options menu and turn the difficulty down to "Story Mode." Completing the 5th and final wave in both arenas should unlock the achievement.
Note: Enemies will use all 3 spell shield variants, so ensure you have a red, yellow and purple spell equip to counter them.