Already some years ago, but I like my stay in Karpathos so much I write a first review anyway.
Alex Hotel is very simple, cheap, but friendly and well situated. The last is not unimportant in Greece. The hotel is near the beach, close to town but not in it. This means its quiet at night. There is a big clean pool and the rooms are small but allright. You will not be staying in your room in Karpathos, so I did not mind. The breakfast is good but also simple. Could be a bit better, but in Greece, it is about lunch, not breakfast.
The owners are greek and friendly. They speak amazingly good English on this island. From all the islands in Greece I have visited (about 10), this one I liked the best.