Hello.
Yes you can use Astral Strike without exhausting it.
Astral strike requires flanking to be used. Zazu doesnt need to exhaust an attack card if he is attacking a flanked target.
Look at this example with Scarlet.
https://www.kickstarter.com/projects/coolminiornot/arcadia-quest/posts/770980
Astral Strike
Melee Attack 3 Dice
Magic
Attack an Enemy anywhere on the board Close to an ally.
(May be used by a dead Hero. Does not trigger Aggro).
Zazu
Does not exhaust an attack card when attacking a flanked target
(a target with at least one other Enemy Close to it).