Bodrum airport is the closest to the apartment at about 70 mins. Izmir is about 120 mins away.
Hire cars are available at both airports and the roads are very good nowadays. Ask for more detailed driving directions if hiring a car.
Airport transfer companies operate from both airports and provide very cost effective transport if you do not wish to drive. Prices start from about £15pp return. Ask for details of companies as many operate a website and can be booked online.