Change the existing content type of the List (or) SharePoint Library using ClientContext

 private void ChangeContentTypeOrder(ClientContext ctx, List list, string contentTypeName)
        {
            Microsoft.SharePoint.Client.ContentTypeCollection currentCTOrder = list.ContentTypes;
            ctx.Load(currentCTOrder);
            ctx.ExecuteQuery();

            IList<Microsoft.SharePoint.Client.ContentTypeId> reverceOrder = new List<Microsoft.SharePoint.Client.ContentTypeId>();
            foreach (Microsoft.SharePoint.Client.ContentType ct in currentCTOrder)
            {
                if (ct.Name.Equals(contentTypeName))
                {
                    reverceOrder.Add(ct.Id);
                }

            }
            list.RootFolder.UniqueContentTypeOrder = reverceOrder;
            list.RootFolder.Update();
            list.Update();
            ctx.ExecuteQuery();
        }