you can call javascript function on events like oncut and onpaste event and write your logic on this function.
function textchange()
{
document.getelementbyid('txt2').text (or may be .value) = document.getelementbyid('txt1').text ;
}
<asp:textbox id="txt1" runat="server" oncut="textchange();" onpaste="textchange();" xmlns:asp="#unknown">
<asp:textbox id="txt2" runat="server">
</asp:textbox></asp:textbox>