How to keep opened coconut safely?
I would suggest just put some plastic foil around it, so the bugs aren't on it. And don't put it in direct sunlight. Since you have no fridge, maybe search for a 'cooler' place (cellar, cabinet, a place where the sun never comes etc.) to put it.

I myself take out all the meat of the coconut, put plastic foil around it and put it in the fridge. That way, it's good for a couple of days.

Do you have a freezer? You can also put some in there. It will be good for longer than a couple of days (up to a couple of months if you preserved it well). I have no idea how it turns out afterwards. I'm guessing it will be a bit less 'juicy'.

Another way to preserve it, is to dry the coconut. But this will of course damage the flavour.