On the server side (during PostBack) use C#'s
Textbox2.Text = "The text you wish to update";
e.g.:
Textbox2.Text = drp1.Value;
On the client side in Javascript (jQuery):
document.getElementById("Textbox_ClietnUD") = $("#DDL_ClientId option:selected").text();
Note that for this to work you will need the client ID (which is auto generated by the page) of the Textbox and the DropDownList.
Each Control's client id can be accessed (in C# in advance) via the
Control.ClientID Property.
Best,
H