I still think it should be out, and sun will not hurt the tree. not having enough water will.
Keep it from drying out otherwise you will have more porblems

yes tropical or not it sould be getting outdoors time the sun will do it wonders...I guess if in autumn the leaves don't change and fall its tropical .

I know it's been awhile since this thread started but I just ran across it.
The Cherry you are referring to is a Eugenia also known as a Brush Cherry (or Teeny Genie). It is an Australian tropical member of the myrtle family and indeed it is sold as an indoor bonsai. I have one I keep indoors for the winter under lights, mist daily and it does quite well. In the spring - summer I keep it outside.