The key to midding fizz is, as someone already mentioned, managing trades to get a pressure lead early on.
I like to think of it in these terms. In starcraft 2, it is commonly said that to win with zerg, as a general rule, you should always have one more expansion than your opponent, and you do this by expanding early.
With fizz, begin initiating trades as early as level 2, and maintain a health advantage by constantly diving in for trades as your counterpart approaches the wave to last hit.
For the trades, as they approach the wave, dash in with Q. As you land, activate W before your first attack lands. Auto attack them twice, maybe three times if you can get away with it, then E to get out.
Usually once you Q to them, they either start to panic and don't cast anything, or show absolutely no sign of worry and start throwing spells out. Just juke a bit between auto attacks if they don't instantly back off, which will put them in an even worse position, and tilts the trade even more in your favor.
As you E away, occasionally they will try to even out the trade a bit and throw a spell out, if that is the case, you can elect to use the second half of your E to go back to them and apply more pressure, as they just wasted a cooldown.
After 6 it doesn't get much more complicated, just add the ulti into the rotation wherever you see fit.
Of course, the usual laws of midding in general apply, after you do it multiple times, you can get predictable. Be aware that while your E does make escape pretty easy, if baited into a situation just that little bit too risky, it can be fatal. Remain aware of how your lane counterpart changes in behavior from level one onward.
Other than that, make sure to get your farm in whenever you aren't harassing.
You could say that it isn't really "sustaining", but ensuring that your counterpart is even less sustainable than you are. If done correctly, they aren't even "trades" really, because often times you will take no damage other than some auto attacks.
Hope this helps, and sorry for the large response.
Best Answer
I wouldn't say it's a good item... It's crucial.
Lichbane has basically everything that fizz needs: AP, Mana, Movementspeed and the Best Passive an Item could ever have for fizz: On-Hit Bonus Magic damage w that scales with AP and that gets applied instantly with your Q.
There is no Item for fizz that comes even close to this one. You should rush a Sheen almost all the time as a core Item on him. Sometimes even before boots of speed. The Active and Passive stats are just too good and they can help you snowball early.