A lot of Fizz's damage is loaded into his R and his delayed W. Since his E is going to be dodging most of your burst your best bet in that match up is just to play as defensive as possible and wait for team fights. Odds are pretty great you're not going to be out-dueling him unless he makes some serious mistakes.
To mitigate some of his burst I would focus on making sure you have a clear 'out' with your Shunpo. That way when he goes in you can attempt to get out of range of his empowered W (which takes a few seconds to fully charge after you've been hit). And, of course, you're going to have to work on predicting/avoiding the shark. Treat an unseen Fizz in your lane the same way you'd treat an Eve or an ulting Rengar. Stay away from unwarded bushes, play cagey as all heck.
Rushing a
can help for when/if he does tag you with a shark.
won't hurt either you you feel he's getting too many engages on you, proving HP and MR to survive his burst while the passive's range pairs well with Kat's kit anyway.