If anything, the cost for the item would go down over time. You're getting the bonuses for free, where as you'd have to buy more items to get them. At any rate RoA first is superior to Archangles.

RoA is a terrible item, so bad that i had to make an entire thread dedicated to my rage at it.

In some ways. It depends on what you need.
Archangel's Staff gives MUCH more offensive power than RoA ever will, at the cost of no survivability. In fact, I believe Archangel's gives more than 80 AP if you have the 1000 mana maxed out, and that's not counting the other mana you have.

I would personally grab RoA and then the staff, if I were playing a pure magic caster, like Malz or Veigar.

I get RoA because i love starting with a catalyst. If i don't i find myself backing to heal without even enough gold to buy anything.
In some rare situations i will skip RoA and just get banshee's but 9 out of 10 times it's RoA before i even finish my boots.

I'm a new player and have most of my games played with Ryze. For his particular situation, where his q ability scales off of his max mana, would it still be beneficial for him to build RoA before Tear of the Goddess, or after? I've actually been building the tear first, then going for RoA (then boots, depends on how the game is going, may build boots sooner). Maybe that is too much of a delay, building it after tears and possibly boots...

It's always a choice. Ryze's passive is going to scale off the tear and the ROA, so it's going to be about survivability, which is going to depend on your playstyle.

If you're pushing like a lunatic every chance you get, you're going to want the earlier RoA to give you a better chance at surviving the inevitable ganks.

If you're a cautious player, keep your minion waves back and harass carefully, you can probably get away with the more agressive tear (or even archmage) first.

The best thing you can do is try it out, and see what happens.
If you're dying, you need more hitpoints - Rylai's or RoA.
If you're running out of mana, you need mana, so Tear or RoA.
If you're just getting pushed out of your lane, you need more health regen - so catalyst
If you're not getting the kills, tear into archmage's is the most damage you can do.
If you *are* getting kills, consider a mejai's.

There's no one best thing for all players or all games or all heros. There *are* bad things that you should basically never get, but RoA is definitely not one of those.