If I move a unit to an empty battlefield, but before I can conquer the opponent plays a lethal damaging spell, can I then play Arcane Shift on the unit and banish it, then play it back to the empty battlefield it attacked?

No, you cannot use Arcane Shift in this situation.

According to the card text and established rulings, Arcane Shift requires you to target both a friendly unit AND an enemy unit at a battlefield to be played. If you have moved your unit to an empty battlefield, there are no enemy units present at that location for you to target.

As stated in FAQ #592, "Arcane Shift requires targeting both a friendly unit AND an enemy unit at a battlefield - you cannot play it without valid targets for both." While you can play [Action] spells during a showdown, you must still satisfy all targeting requirements of the spell at the time it is played.

Because there is no enemy unit at the empty battlefield to target, you cannot initiate the spell.