According to this question: TextBox.ScrollToEnd doesn’t work when the TextBox is in a non-active tab
You have to focus the text box, update the caret position and then scroll to end:
Status.Focus();
Status.CaretIndex = Status.Text.Length;
Status.ScrollToEnd();
EDIT
Example TextBox:
<TextBox TextWrapping="Wrap" VerticalScrollBarVisibility="Auto"
AcceptsReturn="True" Name="textBox"/>