Select 1px aliased border
When I choose Select > Modify > Border and select a 1px border - the border I get is anti-aliased, hence if I fill my selection a 3px border is affected.
Is there a way to select a 1px aliased border?
Instead of going your route and 'painting' the border, why not try layer styles > stroke (outside 1px)?
This should give you a clean 1px border with no alias. Also you can always go back and change its colour/size etc any time you need.
You can access layer styles from the bottom of the layes panel (FX) > Stroke.
Or similar to your approach, you can try edit > stroke (1px) this will 'paint' the border without anti-aliased.