From all achievements in World of Warcraft only a small percent are available to be achieved on the Wandering Isle. Achievement available to be done on the Isle is achievement that requires your character to stay at the island all the time. There are 27 available achievements yielding a total of 270 points. Achievement with "account-wide" progress are not listed as they don't show in the armory when "show only character's achievements to others" option in interface is checked.
1. Level 10 [10p]
2. Level 20 [10p]
3. Level 30 [10p]
4. Level 40 [10p]
5. Level 50 [10p]
6. Level 60 [10p]
7. Level 70 [10p]
8. Level 80 [10p]
9. Level 85 [10p]
10. Level 90 [10p]
10b. Level 100 [10p]
11. Going Down? [10p] - You have to fall from 65 yd and survive. Jumping out from the bridge on higher level of the Isle to the ground below is the best way to do this achievement.
12. 50 quests completed [10p] - You have to complete whole quest chain, so it's a choice between this or Speed of Huo.
13. A simple Re-quest [10p] - Complete a daily quest every day for 5 consecutive days. Turn-in the account wide battle pet daily quests.
14. 5 Daily Quests completed [10p]
15. 50 Daily Quests completed [10p]
16. 200 Daily Quests completed [10p]
17. 500 Daily Quests completed [10p]
18. 1000 Daily Quests completed [10p]
19. Got my mind on my money [10p] - Loot 100 gold from monsters. Long and tedious grind.
21. To all the squirrels I once caressed [10p] - You or someone else has to summon their battle pets representing the pets needed to be /love'd and you use the command on the pets. It will work.
22. Bread Winner [10p] - get 10.000 gold from quest rewards.
23. Justly Rewarded [10p] - get 30.000 gold from quest rewards
24. Love Magnet / Beacon of Love / They Really Love Me! [10-30p] - Get two other characters to use True Love Prism toy on You until you get achievements.
25. The Danger Zone [10p] - Destroy someone's else Crashin' Trashin' Flyer (toy).
Maximum possible achievement points to obtain on a Pandaren Stayer is 270p.