Can I cast a Hero's Downfall after my opponent casts a planeswalker but before he activates it's ability?
That depends on whether your opponent gives you the opportunity. Any time a spell or ability resolves, the active player (whoever’s turn it is) is the first player to get priority (the chance to do things).
So in your example, after the planeswalker enters the battlefield (assuming it’s your opponent’s turn, which it probably is if they’re casting a planeswalker), your opponent gets priority again. If they choose, they can immediately activate one of the planeswalker’s abilities (again, assuming it’s their main phase and there’s nothing else on the stack). Once they do that, you can cast Hero’s Downfall in response to the ability — but at that point, the ability is already on the stack. Hero’s Downfall will destroy the ‘walker, but then the ability will still resolve.
On the other hand, let’s say your opponent doesn’t immediately activate a planeswalker ability, and instead announces they’re going to move to the next phase of their turn (or cast another spell, or activate an ability of another card). In that situation, you’d get priority in response to that action, so you’d be able to cast that Hero’s Downfall before they used any of their planeswalker’s abilities.
Priority is a very complicated topic (as is the stack in general), so if this doesn’t make 100% sense, feel free to ask any followup questions you need to. I also recommend checking out my older tutorials, specifically:
MTG Tutorials #1: What Is The Stack?
MTG Tutorials #2: All About Activated Abilities
MTG Tutorials #4: What Is Priority?
You can also find a list of all of the tutorials I’ve done here. Most of the fundamentals of the game rules are covered there, along with keywords from the core sets, Return to Ravnica, Gatecrash, Dragon’s Maze and Theros.