Punya or ada
So far in this course I have only found "punya", to have but when I was in South Sulawesi some years back the general use was "ada". Like "ada Nasi?" "Ada". This was how people spoke. Maybe not highly grammatical but who wants to spent years diving into the intricasies of university style grammar when you just need to get by without sounding like a toff?
You're right that 'ada' is often used like that in everyday speech.
Please note that in your example, 'ada' is used to indicate the "presence" (not the "possession"):
"Ada nasi? , Ada"
"Is there rice? , There is"
In such a situation, it would be strange to use 'punya'.
'punya'is used to indicate "possession".
'ada' can be used to indicate different things :
"presence, location, possession and used as an intensifier."