Which champion is a good partner for Ryze?

ryze + xin = gg

Blitz works pretty well

basically any decent melee that can do some damage. Xin is awesome b/c ryze's short rune prison stun is all he needs to knock somebody up and explode them. Alistar works with pretty much anybody, and Ali can roam a bit to give ryze the solo for a little while, which could work out nicely

Thanks for the feedback!

Is it just anyone with a stun, gap-closing, or slowing ability?

those champs are just generally good duo laners, ryze specifically needs somebody who can do a good deal of burst during rune prison if he's gonna get any kills in lane, and probably someone kind of tanky to make sure he doesn't get dove.

For example, amumu has a gap closer and a stun, but would not work well because the lane wouldn't have enough damage to kill anyone easily. Nasus really has neither, since wither isn't that great in the lane phase because people usually aren't overextended enough for the slow to increase and for it to be useful, but despite that, he's still great with ryze because rune prison allows him to put out a lot of damage with E and Q that he otherwise wouldn't be able to pull off.

EDIT: that part about nasus wither is probably controversial so ima clarify. Wither doesn't immediately slow enough to allow you to put out very much damage, and when its slow increases, the enemy champ is already under their tower or their lanemate has CC'd you or something while you were chasing the withered champ. Wither is more a means to get your lanemate to stun the enemy champ so you can THEN do damage. The same logic applies to most slows, other than Tryndamere's mocking shout, Nunu's ice balls, and a few others when leveled.

you might consider garen as a viable option...
though lategame he isn't very powerfull, with a strong ally he can push pretty much everyone out of xp range early game.

The best experience I had playing Ryze (who was my main at one point) was with Singed... ZOMFG he would just toss people in back of him, I would snare and burst, he would then goo them as they try to run out of his poison cloud, and then..... RIP [insert name of whoever got tossed by singed].

in my best expieriances with ryze i prefer to have a ranged or melee nuker such as brand becuz of his stun with sear and incredible combo damge becuz with a rune prisoned person thay cant dodge his abilitys or some one like xin or garen.