I wouldn't be inclined to use anything wet: probably you should be very gentle: if the spots are on the outside, you might want to start with a soft brush to get rid of as much of that old mildew or whatnot first, then I'd see if it comes away with a damp q-tip, just dab it away at first. Your big worries are about getting the stuff wet or it drying out with stuff like detergents or alcohol.
It's probably made of paper on cloth or something like canvas. It may be somewhat brittle. The outside layer, you could treat like it's old painted paper and use about the care you'd expect.
Is the mildew damage very old or more recent?
I would not use bathroom cleaner, that's likely too harsh.