I rent this place and other than that, it’s a beautiful older home.
Hi Beth. I cleaned my white porcelain sink with Cerama Bryte and it was amazing. You can peek at it here if you like: https://www.exquisitelyunremarkable.com/2018/06/best-way-to-clean-a-white-porcelain-sink.html
Try soft scrub.
I use Barkeeper's Friend spray. I have a porcelain tub, but the finish is gone and it grabs stains.
I use a soap scum remover such as scrubbing bubbles or lysol power bath. Spray it on, let sit for several minutes, and then use a nylon scrubby to gently scrub the surface. Rinse.
If its rust, I use CLR, Bar Keeper's Friend, or Iron Out. If its sincerely grimey, you can use Easy Off Fume Free Oven Cleaner and a magic eraser to get year's of grime off. I have also cleaned a lot of stained tubs with Dollar Tree Awesome Orange in the half gallon, pour it around the sides, let site a few minutes and then scrub off with a brush or scrubby. Once you get it clean, treat it with Gel Gloss to refinish and shine, or use a carnuba auto wax product and buff with soft cloth.... careful it will be slippery, so add a mat or some of those stick on strips on the bottom.