I'm not sure I completely understand what his "Blood Well" is for. As far as I know once Aatrox gets to critical health he'll kneel down becoming invulnerable and then get temporary health (from the Blood Well). I know that the Blood Well has other purposes too though.
So, Does Aatrox's Blood Well act similar to how mana would on a mage?
No.
There are four general groups of champions in League of Legends right now for classifying how they cast spells: Mana-based, Energy-based, HP-based, and Manaless.
Mana-based champions expend Mana, a secondary bar, to cast spells. Mana can be increased by various items, runes, and masteries. Mana regeneration may also be increased. Mana-based champions have a base amount of total Mana and Mana regeneration, and they tend to gain more total Mana with levels.
Energy-based champions expend Energy, a secondary bar, to cast spells. Unlike Mana, total Energy and Energy regeneration cannot be increased by items aside from the Ichor of Illumination on Twisted Treeline, which, like the blue buff, increase only regeneration. They can, however, be increased by runes. Energy regenerates much more quickly than Mana, at a base rate of 10 Energy per second. Most Energy-based champions also have skills that allow them to partially recuperate the cost of a spell.
HP-based champions like Aatrox expend some of their Health Points to cast spells. They tend to have no secondary bar (although Aatrox is one that does), and the health costs may be flat or, more often, a percentage based on their current or total health. HP-based champions cannot die by casting a spell.
Manaless champions do not expend any resources to cast spells. They are limited merely by their cooldowns. They may have additional bars (e.g. Fury) that have various effects.
Aatrox's passive is Blood Well. It's a revival passive, much like Anivia's. It has a 225 second cooldown. Whenever Aatrox expends HP by casting a spell, it gets added to his secondary bar, which is also named his Blood Well.
If his Blood Well is active (i.e. not on cooldown; it will be red) and he "dies", he will heal based on how much he has stored in his Blood Well. This isn't temporary health though; it's a heal. I am not sure if this is affected by ignite; that probably should be tested. I also am not sure if it will drop buffs and debuffs, though if it's like Anivia's passive, it will only drop buffs.
Blood Well currently has only one other purpose: for every 2% that his Blood Well is full, his Attack Speed increases by 1%.
