I asked a similar question recently. I needed to clean a tub that had old rust stains, very, very bad rust stains.
I had tried Soft Scrub, that Scrubbing Bubbles cleaner, Kaboom, etc.
Nothing would touch it. So I came on here and asked. Someone said Super Iron Out so I bought that, tried it, didn't touch it.
Someone said The Works Toilet Bowl Cleaner (Wal-Mart), not the Tub & Shower Cleaner. So I tried that. Oh. My. Goodness. It "ate" the rust and I just wiped it away. Awesome!!
The tub I needed to clean was old too, like you said yours is. I'm not sure how many years old but it was in the main bathroom when the house was remodeled in 1974 and they put it upstairs. So it is more than 35 years old, maybe 50? I'm not sure.
But it looks so nice now with those nasty rust stains gone!
Give it a try, I bet you'll get good results!