You are right that public hospitals here are quite good such as Hamad Hospital. However, from what I have learnt from colleagues, one needs to wait quite a while there before his turn comes. For other private clinics you don't need to have an appointment. Just walk in pay the fee and wait to be called in. However, many clinics do not have doctors for some types of problems such as yours which I believe is related to the heart. You can get details of private clinics/hospitals in Doha from Google. You can then call them up and find out for yourself if the doctor you are looking for is available with them.