Apparatus, and an associated method, for a PoC (Push-to-Talk over Cellular)-capable communication system. A user equipment PoC box is provided, and a corresponding network PoC box is also provided. A transport mechanism is defined and provided by which to communicate a PoC media burst that is delivered to a PoC box and stored thereat. Control functionality is provided by which to control disposition of the media burst, such as to make notification of a delivered media burst or to dispose of the media burst when no longer needed, is further provided.