"No, I do not know the ticket price." I know perhaps the author wants to highlight the genative case, but if a phrase is in common usage, why is it invalid?
This sentence does sound a little less common to me, but not wrong. I'd report it.
the ticket's price, так не говорят?
Why I cannot say:... a price of the ticket?
Just because that never makes sense in English. We'd never say "a price", I guess because you only pay one price.
"I don't know the price of the ticket" is not accepted