Don't be afraid to use Overload to last hit minions as the spell has a low mana cost and cooldown.
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 aswell, 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.
Muramana can provide more damage (as it procs on autoattacks and single target spells), because of Ryze's mana pool: (250 base + (55 per level*18 = 990 at level 18) = 1240 + 1000 mana due to Tear = 2240 mana * .06 = 134 bonus physical damage per autoattack and single target spell. Archangel's staff only provides (50+(2200*.03))*.4 = +46 damage on Ryze's Overload, his main source of ability damage.
Although Ryze's spells deal splash damage during his ultimate, Muramana's toggle will still proc on the actual target of the spell; the Muramana damage is not splashed.
However, the bonus damage from Muramana is physical and will not benefit from Ryze's MR reduction and penetration.
Consider building Iceborn Gauntlet in combination with Muramana as it will allow to follow up with a powerful autoattack with an AOE slow after casting a spell.