how to pass value from input:text to label with jquery

Is the button causing a full page post? Or are you using an update panel?

As you know text that is set using JavaScript isn't preserved through full page posts. So if you're set on using JavaScript, you're going to have to do the following:

1. Change the update_lbl to an asp:TextBox.
2. Set the readonly attribute to true.
3. Change the client click code on the button: OnClientClick="setText();"
3. Change your jQuery to this:
function setText() {
$('#<%=update_lbl.ClientID%>').val($('#datepicker_value').val());
}
4. On the server side, in the button click event request the labels text. The text has been set by JavaScript, but we have to make the ViewState know that it needs to be a different value:
Private Sub BTN_confirmation_Click(ByVal sender As Object, ByVal e As System.EventArgs) Handles BTN_confirmation.Click
Me.update_lbl.Text = Request.Form(update_lbl.UniqueID)
End Sub

And that will preserve the value that has been changed in JavaScript through a postback. You'll just have to make that textbox look like a label, which isn't difficult.

(OPTION 2) Of course, it would be much simpler if you could get the date from the datepicker_value input. Then you could cut out all of the client side code, leave your label a label, and do this in your button click event on the server side:

Private Sub BTN_confirmation_Click(ByVal sender As Object, ByVal e As System.EventArgs) Handles BTN_confirmation.Click
Me.update_lbl.Text = Request.Form(datepicker_value.UniqueID)
End Sub

This is much cleaner. It may not be possible, however, because your date picker needs to have the following attribute added: runat="server", and that attribute may cause problems for the rest of the page. Or it might be a third party object and you don't have the option to change it.

Hope that helps!

Whoops! What I said about Option 2 was wrong! You don't have to add the runat="server" attribute, you just need to add a name.

<input type="text" id="datepicker_value" name="datepicker_value" style="border:none; " />

Then in your button click:

Me.update_lbl.Text = Request.Form("datepicker_value")