As per my understanding you want to hide your calling number so it will not be displayed on others telephone sets, additionally you do not want to be charged by the Public telephone company.
Sorry, but theoritically it is not possible to hide the outgoing calls from the PTT (Public, local phone company), however there is some ways to hide it from the PABX (old PABXs only)…. regarding the first part, it is only possible to hide the caller ID through the public exchange, if in your country they are not allowing it (as per my one), you can not do it.