I have moved into a new apartment, the bath tub has few rusty spots.

https://www.hunker.com/13423038/how-to-clean-rust-stains-from-a-bathtub

Try CLR, but check with your landlord first, and take a few pictures, so it shows they were there when you moved in. EJL.

New apartment as in a new build ? , the tub should not be worn out yet, but the damage could be from when it was installed. It may be that the spots are rust that has eaten all the way through the finish on the tub, not just stain spots. Hopefully not, but I agree with Ebbjdl, and take pictures showing the date so that you have some evidence to show the landlord.

Your home improvement store has a porcelain repair kit for $ 20 to $30

You can easily DIY

Ask the landlord to repair.

try a magic eraser!!!! It does wonders!!!