The house we rented was the 400 year old former kitchen wing of a country house, complete with those 18" thick stone walls and horsehair & lime render. It was quaint and had a special charm, but the day-to-day living was not the most pleasant. Unlevel floors, cold and damp walls, draughty windows and thresholds, and yet, poor ventilation in the kitchen and bathroom. Mould was a constant problem... one we didn't resolve so much as try to cope with.
In warmer months, we followed the rule of "if it's dry enough to hang your laundry outside, open the windows". But in winter, you don't want the windows open! We didn't like to run the heat because the thermostat wasn't capable of keeping all the rooms comfortable, so we only used the heat manually, turning it on until the room we were using was comfortable, then turning it off again. We probably used it for about an hour each evening, so we could cook and eat dinner without freezing.
We did eventually get a dehumidifier... once we had that, we kept it on pretty much 24/7. It made a big difference in maintaining the air quality. But the mould had already taken hold, so any parts of the house we couldn't ventilate properly were just lost to us.
Oh! And we thoroughly cleaned and painted the bedroom and the bathroom with mould resistant paint about a year before we found our house. With the dehumidifier going, it was enough to keep the mould from coming back. When we were doing the final cleaning at move-out, we didn't have to do anything in those two rooms. (We did have to clean and paint the hallway, kitchen, and lounge/dining room, though!)
Anyway, yeah... I recommend mould resistant paint (if you can be bothered to spend your money and do the work in a rented accommodation), and definitely a dehumidifier. It won't magically go away, but it helped us some.
Plus, we got to take our dehumidifier with us to House, and it's been instrumental in helping us dry it out after it had been empty for a number of years.