WebClient will not support it but with HttpWebRequest you can either try to be nice and send a HEAD request if the server supports it or if it doesn't send a normal GET request and just don't download the data:
The HEAD request:
HttpWebRequest request = (HttpWebRequest)System.Net.WebRequest.Create(uri);
request.Method = "HEAD";
HttpWebResponse response = (HttpWebResponse) request.GetResponse();
string disposition = response.Headers["Content-Disposition"];
string filename = disposition.Substring(disposition.IndexOf("filename=") + 10).Replace("\"", "");
response.close();
If the server doesn't support HEAD, send a normal GET request:
HttpWebRequest request = (HttpWebRequest)System.Net.WebRequest.Create(uri);
HttpWebResponse response = (HttpWebResponse) request.GetResponse();
string disposition = response.Headers["Content-Disposition"];
string filename = disposition.Substring(disposition.IndexOf("filename=") + 9).Replace("\"", "");
response.close();