I have an old steel framed bike (Team Marin) that is getting some rust on the chainstays where the paint has rubbed off – see pic.

The frame is about 1990 vintage, has virtually no value except sentimental, being my first mountain bike.

I don’t want to spend much on it, ie: not a strip and re-paint. What can I do to preserve it?

It’ll probably be fine for several years, but if you really are worried why not just touch up the area with some primer for now?

Rub it down to shiny metal, including the edge of the paint around the rubbed area as the rust will have crept underneath the edge of the paint. Then put a coat of primer and a coat of paint on it, job done. It looks to be silver so you can get Hammerite in about the right colour from Halfords or similar.

That looks just like surface rust and will be pretty much just cosmetic,
rub some hard wax over it to preserve the appearance and go ‘Rat Look’.