Where there is evidence of water problems in your ceiling, what is directly above it? Any water pipes (either supply or waste pipes)? Try and marry up water damage with a potential source directly above, if possible.
I think hold off for a bit and keep an eye on it for the next couple of weeks. Because it's such dry weather, if it gets worse without rain then it's likely not rain damage, it could be a water pipe above. If you want to confirm, look into fluorescent water tracing dye, which you dilute in water and pour down the drain. If you see the dye coming through the paint in the ceiling (with a UV light is better) then it's a piping problem not waterproofing. Any stains on the ceiling from the dye isn't a problem, you'll be painting anyway. You may even discover another issue spot that isn't visible to the naked eye (yet).
DIY waterproofing isn't a hard job, but choose the right one. Most don't like heavy foot traffic, some don't like UV exposure and alot don't like water ponding on top of it (accelerates degradation). It's the best time of year to do it anyway. Brands like Kangaroo or Sika are popular in HK, a 20L bucket is about $1k and should be good for a few coats of non-trafficed areas of your roof.
If it were my place, I'd start with:
- clean drains
- pressure wash the whole roof so you can see the surface better. Most handymen would have one, if not a painter. Need to clean the surface prior to applying any sealants/waterproofing anyway otherwise it'll peel off like skin on a soup
- check surface for cracks or damaged paint/sealants, check water pipes and seals for leaks
- soak the roof and look for spots water ponds. Ponding is never good for any waterproofing membrane
- consider a paint on waterproofing membrane (sika/kangaroo as above) around low points, wall-to-floor joints, around drains
- consider a waterproof tile grout and reapply if existing grout is old/chipped
Now's the time to get it right, the weather is great for it. Much more of a pain when it starts raining and you realise you missed a spot!