There's an indirect but easier way to retrieve the content of the textblock.
In the click event, you can retrieve the object your model by using the DataContext
property:
private void Copy_Click(object sender, RoutedEventArgs e)
{
var model = (Message)((FrameworkElement)sender).DataContext;
// Display model.Text in your TextBlock
}
Just replace Message
by the type of the objects you've assigned to the ItemsSource of your listbox.