Anamur itself is a modern town and not interesting except for the nearby Anemurion, the Anamur Castle (Mamure Castle seems to be a better name) and a decent but not very inviting beach at Iskele, a kind of suburb. The fine Softa Castle is also within easy reach (20 kilometers, frequent busses).
Anemurion (Greek for Windy Spot) is on Cape Anamur, which is the Southernmost point of the Turkish coast in the Mediterranean Sea. Of Greek origin it was incorporated in the Roman Empire in 72 AD. In 650 the Arabs destroyed it completely. In the 12th and 13th century Armenian kings repopulated it, integrating the Castle 6 kilometres hence.
Roughly one and a half hours drive from Mersin can be found in the lovely town of Anamur, famous for its cool climate and great traditional cuisine. The ruins of Anemorium can be found there, within a few kilometres from the town near the coast. There you can see the famed Turquoise waters that make Turkey one of the best tourist attractions in the world.