Private Driver

That’s quite a drive and will cost
You could take the train

Try this
www.mydaytrip.com

I have used Rind Transfers to book private drivers in Italy and it worked out well. They are a British company and put the request out to local taxi companies in the area where you want a driver. They then send you the bids from those local companies and you choose the one you want.

https://www.findtransfers.com/

Apart from the distance, most villages in Cinque Terre are car free. So not the easiest place to be picked up by a car.

Do you mean Montecarlo Italy ( just east of Lucca) or Monte Carlo Monaco?

Given your spelling I suspect the former, which makes your task much easier to accomplish.

Have you asked your landlord- lodging host about this? Arranging for a pick up from La Spezia would probably be the easiest.

If no more than 3 people traveling, suspect cost would be around 200 euro