How about this using XDocument:
var xml = XDocument.Load("yourfile.xml");
xml.Descendants("dropDownValue")
.Where(e => e.Attribute("caption").Value == "2013" && e.Attribute("key").Value == "2013").Remove();
//your xml as a string again
var stringXml = xml.ToString();
Or this using XmlDocument:
var xmlDoc = new XmlDocument();
xmlDoc.Load("file.xml");
XmlNodeList nodes = xmlDoc.SelectNodes("//dropDownValue[@caption='2013' and @key='2013']");
foreach (XmlNode node in nodes)
node.ParentNode.RemoveChild(node);
//your xml as a string again
var xmlDocString = xmlDoc.OuterXml;