Could this also be "I have no cat"?
That would be Níl aon chat agam.
Like Irish, we have a LOT of dialects and idioms. I think neither Duo, nor many of the English speakers account for that.
Is there any important difference between 'I don't have a cat' and 'I have not a cat?'
"I have not a cat" isn't something a native English speaker would say. They mean the same thing, but definitely use "I don't have a cat."
I thought it sound like coat...hmmm
You werent alone with that one, I heard coat too.
I tried saying "a cat is not at me" which would be the literal translation but was marked wrong
It's marked wrong because when an Irish speakers says Níl cat agam they mean *I don't have a cat", they don't mean "a cat is not at me".