Archangel's or Manamune on Ryze?
For the purposes of this analysis, I'll assume that you build Frozen Heart and Rod of Ages on Ryze, in addition to the Mana Charge item. This would cost 7900 to 8500 gold. (Another common Mana item built by Ryzes is Banshee's Veil, increasing the cost to 10400 to 11000 gold.)
With these items and the Expanded Mind mastery, Ryze will have 3362 total Mana at level 18. Assuming a standard 9/0/21 AP mastery page, you'll have 98 AP.
Let's do some calculations now.
With a Seraph's Embrace, your AP will change to 262.1. Q (Overload) will do 490 damage, W (Rune Prison) will do 513 damage, and E (Spell Flux) will do 256 damage per bounce. There are two common Ryze spell rotations: QWEQ and QWQEQRQ. The first will deal 2007 (unmitigated, i.e. assuming 0 MR) damage to a target if his E hits twice. The second will do 2988 damage to each target that E hits twice.
With a Muramana, your AP will stay as 98. However, you'll do bonus damage on each spell. These calculations are a little more complex, but they come out to 2247 damage on the first rotation and 3433 damage on the second rotation. In addition, Muramana is 600 gold cheaper. Note, however, that to simplify things somewhat, this ignores mana regeneration, meaning that you actually deal slightly (though probably negligibly) more damage than these numbers indicate.
Therefore, from a pure damage standpoint, Muramana is slightly superior, though as hammythepig mentioned, Seraph's Embrace has an extremely attractive defensive component.
3.5 Balance Update: Now that Muramana does physical damage instead of magic damage, it becomes slightly less viable on Ryze, since it no longer benefits from any magic penetration. On the other hand, it allows for strong mixed damage from Ryze, which can be useful in AP-heavy team compositions.
Well looking at Seraph's Embrace vs Muramana, the only difference you did not mention was the Mana Regen. Seraph's gives +10, while Muramana gives +7. Ryze could use mana regen, so that's a point towards Seraph's.
As for AP vs AD, It would be better to grab Seraph's and go pure AP, rather than some mixed damage. Ryze is primarily an AP caster, and the team needs him to focus on AP to get past the enemy's armour, that is the whole reason you have AP carries and AD carries, to force your opponents to buy MR and Armor.
Although the single target buff would be helpful, the mana shield can be very nice too. It is especially nice when you can activate it before a team fight, when your mana would be full and you can have a nice hefty sheild. This would be useful also if the enemy targets you, because you are a big AP threat, it would allow you to take a few more blows.
All in all, taking Archangel's Staff seems like the best item of the two for Ryze.
You have to also take into consideration the 3% mana damage on the Muramana toggle ability, that 3% can add up, but is only effective on Q and W (since E is not single target).
There is a nice write up at http://www.team-dignitas.org/articles/blogs/League-of-Legends/2933/Ryze-Muramana-or-Seraphs-Embrace/, the conclusion sums it up though:
Conclusion
Muramana can double your ratios on your Q or W, while Seraph's Embrace can almost double your health. Both have their time and places, but Muramana seems to be better because of the insane damage it gives, plus at a cheaper cost.