If you enter in showdown and you play Stupefy, when do you draw a card: after combat resolves, or on the action of Stupefy?

Yes — you draw as part of Stupefy's own resolution, the moment the spell resolves, not after combat.

Stupefy's ability is "Give a unit -1 might this turn, to a minimum of 1 might. Draw 1" — both effects execute together in written order when the chain item resolves (Rule 340.1), so the card is in your hand before the Showdown closes and before the Combat Damage Step.

Timing-wise: you play it during the Showdown (it's Reaction, so it can go on the chain at Reaction timing), it sits on the chain, and if your opponent passes, it resolves and you draw right then. The sooner you resolve it, the sooner you draw — which matters because the new card can be played later in that same Showdown/chain if it's Action or Reaction.