"Go suck yer blade!"
Can I battle that enemy unit at this turn or should I go to the "move" position?
You definitely need a MOVERE action. So the answer is NO.
Can the enemy also chose to battle?
If you mean "battle right after you place the unit using a MILITIA action", NO. He would need a MOVERE action too. The only way to initiate a battle in Antike II is using a MOVERE action.