It looks human, feels human, can walk and run like us and can even argue with us, but it do not possess DNA. Is it possible to create this homunculus?
DNA stores information to build proteins necessary for body function. You need some information for basic biological function and robots firmware, and thus way to store it. If you want to make robot biological, I assume you don't want it to have mechanical/electronic memory either. If it wont use DNA for it either, you need to make up some new for of memory that will take on this function.
Currently all biological 'living' materials that we are aware of have DNA/RNA. So if you want to cover your robot in 'flesh' that isn't some kind of rubber/plastic/neoprene etc. then either you have to invent an alternative to DNA or it will have DNA. It of course does not have to be human DNA that you use but some kind of DNA would be simplest if you want a 'biological' robot.
TL;DR: Yes, as RNA can perform any function that DNA can.
I'm not sure why you would want to though, as DNA is used by cells as it is much more stable than RNA, unless that instability is the entire point.