Can rising damp first show up as mold patches on a ceiling?

I vert much doubt you have any rising damp , personally i would be looking at other places for water ingress roof, gutter, windows to name a few, according to latest research rising damp is misdiagnosed in over 75% of cases raising up to 95% in older property's, try reading the rising damp myth

The word rising Damp is as it says you are correct you may find if you have a concrete floor installed this may be the problem If it’s a wood floor check you air ventilation ne air bricks injection required

Sounds like a condensation issue to me..black spot mould isn’t associated with rising damp. Try improving your ventilation to the affected area.

From your description, it sounds like condensation/Mould issue you are having. Condensation occurs when warm air collides with cold surfaces, or when there's too much humidity in your home. When this moisture-packed warm air comes into contact with chilly surface, it cools down quickly and releases the water, which turns into liquid droplets on the cold surface. Adequate ventilation is the only way of preventing condensation. In order to reduce condensation you must reduce the excess moisture in your property. Please request a Damp/Mould Investigation from PCA qualified person to determine the best long solution for your property.

The short answer is No. It is always possible to have rising damp as well but that would be unlucky and it is more likely it is all the same problem. Specifically re the ceiling, the mould is telling you the ceiling is the coldest place in the room and, if it is top floor, as I suspect, the first thing is to check the insulation. Warming up the ceiling is the right thing to do. However, if the other problems are also condensation related, it will only create more of a problem on the walls which are now the coldest place. You can try more and constant heating but your description suggests you will need to add ventilation as well. it really needs a proper assessment but I hope that helps as a starting point