If you have 4 days and you love sun, sea, hot, swimming, snorkelling, visiting ruines... I suggest you book a boat for 4 days from Olimpos to Fethiye in the south of Turkey; I did it this summer and it was really fantastic... the crew was very nice, I met charming peopl...
Istanbul to Amasra, only the way you can go by bus. There are no train and flight. From Istanbul you should take a bus to Karabuk, which is a bigger city in this area. As soon as posibble you arrive in Karabuk please check a minibus for Amasra. From Istanbul to Karabuk takes 6 hours and Karabuk to Amasra 1,5 hours.