"All Up Round" means the missile is delivered as a completely assembled weapon, except perhaps for quickly attached components such as wings and fins. For example the AIM-9M (in USAF use) is delivered from the factory in a four-round container as a completely assembled missile except for the wings, fins, and TMU-72 coolant tank.
Back in the 1960's and 1970's, many weapons needed to be built up from separately packed components. You might receive the rocket motor in one container, the warhead in another, the control surfaces in another, and so on. This required time to unpack everything and assemble the various parts, not to mention the extra storage space required, more handling, accountability, etc.
There were exceptions, of course, such as the AGM-65 Maverick and (probably) Naval missiles such as Standard, I'm guessing.