Unique Passive – Insight: Gain ability power equal to 3% of your maximum mana. Unique Passive – Mana Charge: Grants +8 / 10 maximum mana (max +750 mana) for each spell cast and mana expenditure (occurs up to 2 times every 8 / 6 seconds).
By building a full set of items giving mana, (fully stacked Archangel's Staff, maxed Rod of Ages, Iceborn Gauntlet, Frozen Heart and Lich Bane) you will acquire total bonus 3150 mana, which will grant total 94.5 bonus ability power, not counting for champion's base mana rating and other stats given by those items indefinitely.
Archangel's Staff has great synergy with a Rabadon's Deathcap, as it increases ability power by 30%. This means that whenever your mana is increased, the Archangel's Staff will increase your ability power, and then the Rabadon's Deathcap will increase it again due to the 30% ability power passive.
To gain more AP from the Archangel's Staff than a single Rabadon's Deathcap (which gives 156 ability power if alone) a champion would need more than 3200 mana. However, a champion would only need 2000 mana for the Archangel's Staff to provide more ability power than a second Rabadon's Deathcap.
With the passive of this item, each 100 gold spent on mana at base efficiency will grant additional ability power worth 32.625 gold.
Note that most champions have around 800-1000 base mana at level 18.
Not counting manaless champions, Vayne and Ashe have 208 mana at level one, the lowest in the game, which combined with the staff's own 250 mana already equals 458. Taking into account the time (and therefore level) it takes to get the gold to buy it, it is practically impossible for this item to be gold inefficient.
Archangel's Staff grants 1 ability power per ~33 mana, giving a total of 67.5 ability power at zero stacks. It gives as much ability power per gold as a Needlessly Large Rod at 2500 mana.