Opponent has a 5 might unit alone at a Forbidding Waste. I enter with a unit; cast Arcane Shift targeting my unit (replaying it to base) and my opponent's unit. Does his unit die?

To answer your question: Yes, the opponent's unit dies.

1. Targeting and Playing Arcane Shift

• Targeting: Arcane Shift requires targeting both a friendly unit and an enemy unit at a battlefield to be played (see RiftJudge FAQ #378, #10139).
• Legality: Your unit and your opponent's unit are both at the battlefield. Because Arcane Shift is an [Action] spell, it can be played during a showdown per its card text ("Play on your turn or in showdowns").

2. Resolution and Might Reduction

• State: The opponent's 5-Might unit is currently defending at Forbidding Waste. According to the battlefield's ability, it has -2 Might, resulting in a current Might of 3.
• Resolution: When Arcane Shift resolves:
  1. Your friendly unit is banished and replayed.
  2. The spell deals 3 damage to the opponent's unit.
  3. Because the opponent's unit remains a valid target, the damage is successfully applied.

3. Lethal Damage Check

• According to Rule 710, units are evaluated based on their current Might.
• The unit has 3 current Might and has been dealt 3 damage.
• Rule 323.5.3b states: "All Units that have non-zero Damage marked on them equalling or exceeding their Might are killed."
• Since the damage (3) equals the current Might (3), the unit is considered to have received lethal damage and is killed.

Additional Clarifications

• Targeting Requirements: Per FAQ #378 and #10139, you must have both a friendly unit and an enemy unit to target to play Arcane Shift. You satisfied this requirement by targeting your unit and the opponent's unit at the battlefield.
• Illegal Targets during Resolution: If your unit were to leave the board before the spell resolved, the damage instruction would still attempt to resolve because Arcane Shift is a multi-instruction spell and a single illegal target (the friendly unit) does not invalidate the entire spell (see Rule 359.3.e.8).

Note: I could not fully verify this ruling against the retrieved rules, specifically regarding battlefield control during a showdown.