Can I play Punch first to save a unit in response if my opponent plays Falling Star?

No, you cannot play Punch First in response to Falling Star.

The reason is the difference between [Action] and [Reaction] timing:

1. Falling Star is a standard Spell. When it is played, it starts a Chain.
2. According to Rule 334.1.a.1 and 334.1.a.2, cards cannot be played during a "Closed State" (when a Chain is active) unless they have the [Reaction] keyword.
3. Punch First has the [Action] keyword. Actions can only be played when the Chain is empty (an "Open State").

Because Falling Star is currently on the Chain, the game is in a Closed State. You would only be able to play a card with the [Reaction] tag to save your unit. By the time the Chain empties and you are able to play an [Action] like Punch First, the Falling Star will have already resolved and dealt its damage.