It can be dispatched, and use Exhaust abilities, in the turn it came on board during a watch.
Watch — :Repair 1 damage on a Mech you control.501.6, 801.8
Exhaust it to use it. Choose a damaged Mech you control and remove 1 damage from it (801.8). Its hull never goes above its printed value, and its plating isn't affected. Only in your own watch, with the crossfire empty; it resolves at once. Not in the turn it came on board during a watch, unless it has Quickstart.