I think I understand what you're trying to do... You'll have to call this on the client side, but if you're button is doing a postback, I believe the text on the label will be reset anyway.
function setText() {
$('#<%=update_lbl.ClientID%>').text($('#datepicker_value').val())
}
And in your button, you can TEST like this. This will STOP the postback.
OnClientClick="setText(); return false;"