ok, check it your self
remove the bottle, fill it with water and keep it at dry place at standing position, if its leaking then try to stop the leak with GMSA glue or and other thing, liky magic depoxy, otherwise change it. Its made of quite hard plastic it shouldnt leak like this. May be your radiator is leaking which then sucks water from reservoir bottle.
Rust inside bottle is no problem, its plastic, remove it and wash it with surf using baby feeder brush, then rinse it thoroughly to remove surf and fix it back. I did same about a month ago. and use coolant instead of tap water.