You do not need to beat all the Cups on all Difficulties. Here is a chart of the Difficulties you need to play:
Pizza Cup - All 3 Difficulties
Toys Cup - Only Easy
Fruit Cup - All 3 Difficulties
Hook Cup - All 3 Difficulties
Hot Dog Cup - Only Medium
Royal Cup - All 3 Difficulties
**Hard Difficulty isnt really hard. Just make sure you drift on corners for extra boost
Achievement Breakdown: