Surprisingly, I never came across a repeating decimal, which did not include the last digit, so I'm wondering if this would even be a valid notation?
$$0.\overline{0}1$$
So the following statement would be true?
$$1 - 0.\overline{9} = 0.\overline{0}1$$
Thanks for clarification.
No, it's not valid; if we specify any digits after the repeated sequence, the sequence has to repeat a specific, finite number of times. Also, note that $0.\overline{9} = 1$, so $1 - 0.\overline{9} = 0$.