In order to show screentips for the items of the drop-down popup, you can subscribe to the ScreenTipNeeded event assign a screentip when it is needed and about to be shown:
public Form1()
{
InitializeComponent();
this.radDropDownList1.DropDownListElement.Popup.ScreenTipNeeded += new Telerik.WinControls.ScreenTipNeededEventHandler(Popup_ScreenTipNeeded);
As to the main part of RadDropDownList, it consist of several smaller parts (ArrowButton, DropDownListElement and TextBoxItem) that need to have screentips assigned to them:
RadOffice2007ScreenTipElement arrowButtonScreenTip = new RadOffice2007ScreenTipElement();