Can you use Arcane Shift on your minion if there are no enemy minions in Battlefield as the text reads the 3 dmg as resolution?

No, you cannot use Arcane Shift if there are no enemy units at a battlefield.

According to RiftJudge FAQ #378 and FAQ #8655, Arcane Shift requires you to have valid targets for both parts of the effect—a friendly unit to banish AND an enemy unit to deal 3 damage to—at the time you play the spell. You cannot initiate the spell without satisfying both targeting requirements.

While it is true that if a target becomes invalid after the spell is already on the chain (for example, if the enemy unit is removed by another effect in response), you would simply "do as much as you can" (resolving the blink but not the damage), this does not allow you to bypass the initial requirement to have both targets present when you first play the card.