I fly International Direct to Beijing. I stay 5 days. Then to Hong Kong 3 more days. Works?

You should keep in mind that your flight to Beijing should originate from some other country, different from both mainland China and Hong Kong. So, if your itinerary is X country (not mainland China and Hong Kong) - Beijing - Hong Kong and if there are no other stops that you have not mentioned and that you are of Cypriot citizenship, you are definitely eligible for 144 hours visa free transit in Beijing.