flying_hamster,
You aren't going to find any budget accomodation near Cathay City, if they won't let you stay in the Headland. If you're on a tight financial leash, I can suggest you look at some hostels in Kowloon. Here is one, check it out:
Hong Kong Guesthouse accommodation - Man Hing Lung Guest House, Tsimshatsui, Kowloon, Hong Kong
You will spend quite a bit of money and time commuting between the airport and the hotel, but I think this is one of the most reasonable deals you can get for your buck. Unless you are willing to try Chungking Mansion, that is.
Good luck!