This rust just keeps coming back in the sink. I use rust remover and it works a little while and then the rust returns.
You may need to replace that metal drain piece : https://www.youtube.com/watch?v=iQaDgfc-MYI
It might clean up and really shine using baking soda and vinegar. Otherwise, you have to replace the source of the rust.
My suggestion is to ask your home center for the best Rust Remover, my husband told me a good rust remover is something like Envo rust, ask about that & that is what someone said.
I hope this helps you
Rustoleum makes a wonderful rust remover. Try it. After you remove the rust, Paint with Rustoleum to keep it away.