Well, her mobility doesn't really work that well against Zed. But it also works against her as she can jump all over the place avoiding Zed. edited this part for clarity
I find that the best thing to do in my ELO (high silver) is to play a champion with a silence or hard CC, and just play an extremely passive phase until 6. The most important thing against Katarina is to not let her hit you frequently - she will poke you into the ground. Focus on CS, silence/cc her when she ults, and you should be fine.
Specifically, Malzahar and Ahri work very well. I've seen quite a bit of success with Lux too for some reason - but that doesn't shut down the ult, just combos all her health away.