Does Public Execution target the enemy unit on cast or resolution?

On cast. Public Execution chooses (targets) both the friendly unit and the enemy unit when it's finalized onto the chain — not on resolution.

"Choose a friendly unit" targets the friendly unit, and "Kill an enemy unit with less Might than it" targets the enemy unit, with "with less Might than it" acting as a targeting restriction (Rule 355.10). You must have a legal enemy unit — one with Might strictly less than your friendly unit's current Might — before you can even play the spell.

If the Might math changes before it resolves (your friendly unit loses Might, or the enemy gains Might so it's no longer "less"), the enemy target is no longer legal and is simply unaffected when the spell resolves — you can't swap to a different unit.