From an aesthetic viewpoint, it's as if Gragas and Lee Sin had a ******* child together, but I do like the idea of the drunken master style of combat. That being said, I feel like it could be almost a foil to Lee Sin, whereas Lee Sin was the disciplined star pupil and Fing Shu was the monk in the garden completely smashed. I also feel that there can be more than one drunk character or "fighting monk," but keep in mind that I believe Riot's goal with each champion is to remain individually standalone and only to include it if it really adds something to the game.
One question though: is your champion mana-based or energy based? (Or "alcohol" based, which would be kind of cool)?
Passive: Drunken Stagger
Both from the mechanic and from your comment I get the feeling as if you're really trying to get a dodge mechanic in without it being dodge (which is pretty much what you said). An alternative I might suggest is having something like:
Each time Fing Shu uses an ability, he gains a stack of Drunken Stagger. Each stack of Drunken Stagger grants 10% chance to dodge a single target attack or spell. Lasts 3 seconds.Nerve Strike
Two kinds of disables...hmm. Irelia's E works because one is pretty much a superior form of the other (Stun in all circumstances beats a Slow, I would think), and it's a sort of "risk reward" scenario. This move, however, serves two very different purposes I would think with positioning. This on top of the fact that it's a very short cooldown, which means that it's very easy to keep someone screwed consistently (with level 5 Q, you're capable of keeping a caster silenced for 1/3rd of a fight if you stay on top of them). One thing I think you could do if you increased the cooldown would be to give him a cool Exhaust like feature, with some damage to couple it. That is to say, Does X damage and reduces their damage dealt by 30% for 1.5 seconds.
I'm not a particular fan of this ability. From the description it feels a lot like Varus' arrow but with a Gragas barrel, even though it's a dash. I wish I could be more helpful in giving feedback to particularly what can be done to improve it. I would simply steer clear of anything cask related.
Like The5lacker said, the percentage is too high. Remember that Malphite's is pretty low, and that Runaan's Hurricane is still only 50% of Attack Damage. As for the active, it seems a bit weird to have three different abilities on one key, but I can see it as being similar to Riven's Q which I like. I think also from an aesthetic and mechanics viewpoint, it'd be better if you could only land all three hits if you were to hit the same champion with each application.
Being drunk should be impactful, and I feel that the ultimate should really tie the aesthetic together for the champion. In that sense, give him something pretty substantial, but take away from the hero as well. I played a MOBA game a long while ago in Warcraft 3 which used a drunken monk style champion, and the way they pulled this off was that getting drunk made it so your mouse input always worked with a 90 degree offset (so clicking at 12:00 on the screen to move the character up would make them instead move to the left). You could give him some ridculous bonuses but make it so he only hits every other attack (50% miss rate?) or something along those lines. Just spitballing here but the ult is lacking right now and needs that extra bit.