Jhin does some pretty insane damage and can be really hard to get once he snowballs, but early game, he should be manageable. What kind of sup is laning with you and what kind of sups do you see him usually with? That will make some difference if you are trying to take him out pre-6 to 6, but overall, the key to taking Jhin out is to play around his immobility and reload times. If you can watch his bullet count, then time your poke for when he is reloading. His reload time is a flat 2 seconds which is a long time to do some damage. Also, as Cait, you can further inhibit his movement with your trap placement. Let him push so you can freeze the lane near your turret and zone him/or his sup with your traps to keep them at a distance. Cait's poke should be your best bet for dropping him down because of her range. Then the sure fire way is to call for a gank from your midlaner or jungler (any assassin should 100-0 him).
Another trick that ive started using against Jhin as Cait is that when I see him put down a trap, I also put down a trap on it to mark it. This at least cancels out the benefits if they try to escape through their traps to catch you.
I started playing Jhin as soon as he came out and had some great success with him. As ADC, I will hover him as my first pick, but as soon as I see an assassin on the other team like Zed, Akali, Talon, I switch over to another ADC.