Work done on a container, resulting in pressure change
Yes, the explanation 


  the container does not expand, so as there is no change in volume of the container, no work is done on the container.  


is right as it takes into account that the container is rigid, so its volume can not be changed. Thus, there is no work done by the gas on the container.

Also, if you say


  " pumping gas into the container require work in itself, due to the effort required to get the gas molecules into the container "


Yes, there is work done on the gas, but it is by the piston(or the pump).

You should see how the laws of Thermodynamics make clear about the boundary between system and its surroundings, so work has to be done by "something" on "something".
Also, In the question, it is clearly asked "Which statement is true of the work done on the container?" So, as the container's volume is unaltered, there is no work on it.