Frost, do you actually play Poppy, or are you just looking at the numbers on her passive and thinking, "Wow, so OP, much early danger,"?
First, Poppy loses to every single ranged champion early, no exceptions. A lane bully in that vein (Teemo, Quinn, &c.) will completely ruin her, because she has no way of catching until level 3 + boots and can be kited even then because without items she has very poor damage apart from her Q burst.
Second, she has no reliable damage output at level 1. If you're competent, you take W to help resist early harassment, which will at most give you +15 armor and AD if you and your opponent each get 5 AAs off in quick succession. Or you can pop it, and burn something like 40% of your mana for an early trade. And yes, the passive will help if you get down to really low health in a level 1 fight.
Here's the thing: Why the fuck would anyone all-in a Poppy at level 1 with a champion that has to get close enough for her to hit back? What would happen in a realistic theoretical match would be this: If enemy laner has medium-long range abilities, they harass Poppy without getting in range for her to fight back. If they don't, they come in and use their lower-cost, higher-impact abilities for a brief burst and then back off before Poppy gets to make use of her W or passive. So, for example, a Darius Qs and walks away. An Irelia lets you hit her, then stuns, AAs a time or two, then walks away. A Nasus Qs you, walks away, and sustains back up. A Maokai W+Qs you, walks away, and sustains. A Jax stuns you/a Fiora parrys you and AAs before walking away. A Teemo/Vayne/Quinn/Vlad/Gangplank/Whatever stands in minions, pokes you whenever you come to CS, and backs off whenever you try to trade. &c. ad nausem.
That aside, you're overestimating how much the passive does when you don't have anything to boost your own damage output. Sure, you can take more than most, but it doesn't matter if the enemy laner can avoid getting into extended fights until you're low enough to burst down. That Mundo was pretty much the perfect opponent for a Poppy for reasons which have already been adequately covered:
1. Super-passive play.
2. Low early damage.
3. Let Poppy farm.
But okay, Poppy has a super-strong early game, doesn't need items, and doesn't have mana dependency issues. If you believe that then I've got a bridge to sell you.