A bit like jwf said, I would advise you to stay far away from brushes, that way he can't constantly jump+Q you. Also, it may be a good idea to let yourself get pushed and farm under tower, or forcee him under his own (to remove some trade pressure, since he won't force trades when he has to farm under tower, and he won't be stupid enough ot towerdive a poppy).
Otherwise, you should be able to outtrade him in quick trades and with minion advantage at level 1, 2 (with stun when he goes back to brush), and probably 5/6. After that, it's really about how you choose to play the game. You can choose to recall and get Tabi + Sheen to simply AA+Q him every single chances you have (he can't even remotely outtrade that unless he blows his full rotation, which means you can just AA him continuously until you lash out another Q). You can choose to get boots 1 + Tiamat so that you can more effectively push him under tower (most gankers won't be able to gnak you anyway since you're poppy). You can choose to take cutlass + boots + dagger(s), and simply farm under tower passively (but this lets him roam a lot more, so prepare to force-push whenever you know he left).
As for leveling up, I would still go R>Q>W>E, with E at level 2. I have only 1 runeset for Poppy, which is my support poppy page, but I may actually take a little bit more armor if I had the choice (and drop flat MR).