I've done it to mine and my rental. Just know that it's not a permanent fix. it will get some chips in the paint. that;s why I said make sure the tub re-glazer company does the proper prep. If they don't get off all of the old stuff, the new glaze will not adhere.

Also advise your landlord that it's nasty and grimy and chips come off in the water when you bathe (if you have a small child, even better for you health-wise). Maybe he isn't aware of how bad it is. And he can use it as a tax write-off

A decent landlord of a nice apartment should replace the whole tub. As Beth H. wrote, it's a write off.

Yes...definitely contact the landlord. We have rental properties and it's our responsibility to maintain this type of thing. Sometimes it's a matter of the previous tenant(s) not bringing it to the landlord's attention. However, it should have been taken care of before you moved in.

Looks like they pulled off a sliding door and track. That line is where the lower track was caulked down.