Since nobody else has replied, I'll try to give som advice.

Your options will be limited at the hours you are talking about. Probably you will have to go by taxi or car unless you are willing to spend the night in Izmir.

There is no public transport that I know of directly between the airport and Cesme. There are buses from Balcova and the central bus station in Izmir. If I remember correctly, they take between one and two hours, depending on traffic and where you get on.

You can take a city bus from the airport (domestic terminal only?) to the central bus station, if you can find it. The other public transport from the airport is the Havas shuttle which drops you off downtown outside the Swissotel. To get to the central bus station you then need a taxi or a city bus (number 169, I think). Or you can just take a taxi from the airport to the bus station. In any case, going from the airport to Cesme with public transport is neither easy nor quick, and not really feasible at your travel times. Also, bus and taxi drivers don't speak English. The central bus station is considered by locals to be the place where you are most likely to be pickpocketed in Izmir.

I'm afraid I cant help you with price.