If you haven't you'll need to implement INotifyPropertyChangedINotifyPropertyChanged for your class that you're binding to.
Also, when you say you want the ListBoxListBox item to be updated immediately, you mean that you want it to change as you type in the textboxTextBox. By default the TextBox.TextTextBox.Text property updates it'sits source when it loses focus, but you can change this by setting the binding UpdateSourceTriggerUpdateSourceTrigger to PropertyChangedPropertyChanged:
{Binding Source={...}, Path=Whatever, UpdateSourceTrigger=PropertyChanged}