needs to build mana to begin scaling, and the lack of mana/MR items means you'll innately have an advantage as a magic damage-dealing champion most of the time. He also has a weak early game, so the three champions I'd recommend would be
(you can kite, poke and harass him pretty much for free),
(you'll completely deny his early game and set him back a huge amount) and
(Ryze's spell spam means Mao can spam heals).
Aside from counter-picking, one thing to know is that, early on, you should not back away after Ryze uses his spells on you. Most people do, for fear of taking more damage, but early in the game at least this is an indication that Ryze just blew all of his cooldowns and is at his weakest, allowing you to let rip on him. Ryze has terrible farming capabilities pre-6, and pushing him to his tower will deny him a lot of CS. He also doesn't handle ganks well, and if he's not pushed to his tower will easily die to a gank or be forced to blow a flash.
In the event that Ryze does get fed, ask your AP and/or burst damage dealers to focus him in a fight, as otherwise he'll deal heavy damage to everyone around. Aside from his root, he has poor crowd control and can be easily focused. Grievous Wounds will also help, as Ryze will heal for a lot while his ult's up.
and
are both excellent counter-items to Ryze, and if you're melee
will allow you to deal a lot of damage to him, even if the armor doesn't do much against him specifically.
will heavily mitigate his only instance of CC along with his damage, and getting these early will prevent him from exerting too much control over you.