You don't choose the value of {X} in the activation cost of Elite Arcanist's ability. It's defined by the mana value of the exiled instant card.

The copy is created in and cast from exile. You must still follow any timing restrictions, such as "Cast [this spell] only during combat."

You can't pay any alternative costs such as overload costs. You can pay additional costs such as kicker costs. If the spell has any mandatory additional costs, you must pay those.

If the copied card has {X} in its mana cost, you must choose 0 as its value when casting the copy. Don't confuse that {X} with the {X} in the activation cost of Elite Arcanist's ability.

If the exiled card is a split card, then X is determined by the combined mana cost of both halves. However, when you cast the copy, you choose only one half to cast. You may cast a different half each time you create and cast a copy.

If Elite Arcanist leaves the battlefield, you will no longer be able to copy the exiled card. If it returns to the battlefield, you'll be able to exile another instant card from your hand, but Elite Arcanist will be a different permanent with no connection to the first exiled card.