I have an old white porcelain sink, so I can relate!

I use baking soda just like Ajax or Comet would be used. I was using a "green scrubby", which really got it clean. Now, though, I switched to one clean rag a day and scrub the sink with it, too. As long as I clean it every day, it stays looking clean and white.

If I miss a day, then it takes some extra effort with extra baking soda and a less damp rag (to increase the grit factor).
Two ideas:
~ You can create a fizzy cleaner by shaking baking soda around and then sprinkling white vinegar. It fizzes up and works really well on laundry, so I wonder if that would help your sink?
~ You can add sea salt (or any salt probably) to the baking soda to scrub out the sink. The salt adds some grit without being too harsh.
Best wishes!