Bleach also works, just make up a solution of Domestos or something like that as directed on the bottle. You will need to hold the bits under water for a while as they will float of course. The most aggressive cleaner is caustic soda. This strips off propolis but wear full PPE - goggles, rubber gloves, wellies and plastic apron. The best rubber gloves are the sorts from farm suppliers with long cuffs. If you've painted the Apidea the caustic soda will take the paint off if you leave it in too long.
You can use more or less any sort of paint suitable for outdoor use, except of course anything with cellulose in it, like a car paint. For just an Apidea or two the main issue would be not buying too much. Cuprinol Garden Shades is fine but you may struggle to get it in a small enough tin. There are outdoor emulsion paints and of course various outdoor paints thinned by white spirits. These will work fine but be careful not to bung up the sliding bits in the Apidea, which is why I haven't painted mine.
I painted some poly nucs bought the other week and the recommendation was exterior masonary paint. I did a small test area and it worked fine. I now have some nicely painted poly nucs the same colour as the house !
So based on that I would say it would work on the Apideas.
Also in the instructions it recommends weak bleach or Virkon S to clean the inside.
I assume the the apidea is made of polystyrene (HIPS), generally, this is not suitable for outdoor applications unless it has been UV stabilized. This can be done as part of the manufacturing process or by 'painting'. Does anyone know if Apideas are UV stabilized?
Some suppliers recommend latex based paints, but stay well away from cellulose based.