As you're fighting King Boar, 1 of the people by the 3 buckets will occasionally run over and fill the bucket. Grab the apples before King Boar eats them or he'll regenerate.
If you don't care about extra apples, just kill him quickly and grab the apples that are offered afterward.
If you want extra apples-- up to 4-- try not to kill King Boar right away, just kill the princes, and watch for a person to fill one of the buckets so you can run over and get it before King Boar does. After you've collected 3 buckets that way, you can go ahead and kill King Boar and collect the 4th bucket afterward.
Don't kill the big boar, just kite it. When the smaller boar appears, kill it and then grab the apple when the bucket gets filled. Continue this way, not killing the big boar, but killing the small one, until you have collected 3 apples, then it is okay to kill the big boar and get the fourth apple.
Fan of playing magblade since 2015. (PC NA) Might be joking in comments.
-->(((Cyrodiil)))<--
For each apple you will receive one apple drink. 4 apples->4 drinks
I have wondered if there is anything special about the drink and have not discovered anything special. For instance, it didn't stack with my current drink. That is the only thing I tried that I thought would be wonderful.
Fan of playing magblade since 2015. (PC NA) Might be joking in comments.
-->(((Cyrodiil)))<--
I fight the boar, focusing on the ghosts until I get 2/4. Then I kill the big boar for 3/4 and leave the Arena. Then I go back in and complete the other arena quest for the Breton, kill the boar and grab 4/4. Then I turn both quests in.
You will only need 1 out of 4 apples to complete the quest. Collecting 4 apples gives you 4 of the (very weak) stamina regen drink instead of 1 for the time you've wasted so it's really not worth waiting around for them to spawn.