Don't be afraid to use Overload to last hit minions as the spell has low cooldown and low mana cost.
Chaining abilities in the right order is key to make optimal use of Arcane Mastery . For starters, begin with Overload and use it every 2nd spell afterwards. Advanced combos rely on making use of Arcane Mastery to cooldown priority spells.
When low on health, don't be afraid to use Desperate Power + Spell Flux on a minion wave to regain some health as his ultimate has a low cooldown. Since Overload has a low cooldown as well, use this as much as possible while you ultimate is on cooldown to reduce it using Arcane Mastery.
There is a bug that allows Ryze to lower his cooldowns while farming : While Ryze is about to kill a unit with his Overload for example and throws another spell very shortly before the minion dies, for example, he can use another spell like his Spell Flux before the impact of the Overload so that all of his cooldowns get lowered even though the last spell he threw isn't on cooldown. This bug allows you to get your Desperate Power very quickly in order to shove the lane or do the wraiths, or be able to gank after he pushed the lane. Also, all the other spell cooldowns are lowered by a second, so Overload is way more spammable.
The choice between Manamune and Archangel's Staff depends on the player and want the player wants to be built on Ryze. An Archangel's Staff can be built into a Seraph's Embrace, which provides more defensive stats for Ryze as it expends a certain percentage of mana for a shield (which Ryze should have a lot of mana in his build, which means stronger shield). The Manamune path provides more offensive stats as it amplify Ryze's autoattacks and his spells thanks to Muramana's toggle. It doesn't hurt to build both items at the expense of building two Tear of the Goddess longer.
Avoid building items with the spellsword effects, namely the Trinity Force, Lich Bane and Iceborn Gauntlet, despite they ALL give mana and ability power as Ryze minimal cooldowns and ability spamming leave little room for auto attacks with high CDR, as his cooldowns become so short he can nearly enter a non stop casting barrage by using his passive and proper use of smartcast.
Iceborn Gauntletcould be obtained as it gives a very large amount of mana and useful armor. The basic attack proc isn't all that important (or actually usable) as the item seems a godsend for Ryze given the stats it gives. Unfortunately, the Frozen Heart actually gives superior stats, builds similarly, is cheaper and has an aura. The auto attack proc is not particularly useful as Ryze is rarely attacking by late game.
Trinity Force is more AD oriented and the large majority of its stats like attacks speed or critical change are a full waste on Ryze. On the same vein Ryze, doesn't even have particularly good base AD or an on-hit ability so the procs are pretty lame or barely usable
Lich Banewould be nice as Ryze might end up with a decent ability power pool due to Seraph's Embrace and maybe a magic penetration related item like Void Staff, but once again, he has very little time to actually proc the passive.
Because all of Ryze's abilities are point and click and rely on him entering a certain range, picking champions that outrange him or that can poke him without entering his ability range are good choices.
Stay back when you notice him activate his Desperate Power. It gives his spells AoE damage, causing him to deal a lot of damage when facing him in a minion wave.
Although Ryze is very strong at late game, he has a weak early game. Putting pressure on his lane or denying his chance to farm or kill will reduce his effectiveness in the later stages of the game.
Ryze takes a while to rack up mana via Tear of the Goddess - denying him farm and killing him early on will hamper him in the later stage of the game.
Late game, although Ryze deals a high amount of damage and is quite tanky, his short range (particularly since his recent nerfs) and relatively low base movement speed makes him quite vulnerable to kiting. Beware, however, as the MS buff he receives during Desperate Power can make this tactic temporarily less viable.