If your opponent wants to use Asuryan's Cleansing on Unit A, he must do so in response to you PLAYING Unit A. The wording you quoted from the rule book is saying that during your Capital Phase, you, as active player, can play units, supports, quests, and now Legends (but not your opponent). According to the FAQ, playing these cards during your Capital Phase are each considered a "limited" form of action, in that they start an action chain, but cannot be played in response to anything.
Therefore, when you PLAY Unit A, an action window opens, and if your opponent does not use that time to play Asuryan's Cleansing, he will not be able to play it later on Unit A. If, in your example, Unit A enters play, the opportunity for your opponent to play Asuryan's Cleansing will have passed and he cannot legally play it until you PLAY Unit B or some other card. Remember that there is a difference (or timing gap) between playing a card (spending its costs and placing it in a chain from hand) and it coming into play or resolving.
Last edited by bitva (2013-06-15 20:24:05)