How does this Imperial Assault character 'Shoot First'?
The Shoot First card can be applied to any Imperial group, including villains, and that group will get a single activation through the use of the card, which translates to two actions.

Explanation:

Shoot First reads:


  Keep this card secret. Play it during any mission at the start of a round. An Imperial group may interrupt to perform an activation. Then, discard this card.


The card doesn't care about any special rules about its target in a specific scenario. It just does as written unless explicitly prevented elsewhere by mission rules or by the cannot keyword, which is an exception to the Golden Rules. (Page 2, Rules Reference Guide)

Phrasing which contains a finale spoiler: 


   Instead of activating as normal, Darth Vader performs 1 action after each hero's activation. (Page 41, Campaign Guide)


Even though the mission rules prevent the gain of activations every round as normal, it does not prevent other gains of activations.  It instead reads as a replacement effect for the core rules which govern figures gaining activations during the Activation Phase. (Page 3, Rules Reference Guide)

Therefore, the card can grant one activation.

Activation is defined as: 


  During a figure's activation, it performs up to two actions. (Page 3, Rules Reference Guide)


Thus, the group performs up to two actions.

Group is defined as:


  All figures that correspond to a single Deployment card or Hero sheet are a group. (Page 13, Rules Reference Guide)


So this works on both regular deployment groups and villains.