Yes, it is possible but anode should be sodiated as Na-ion source.

Dear Dr. Pandey .. In most of the rechargeable batteries the cathode active material contain ions and the charging will cause these ions to move from the cathode to the anode. Therefore, this ion transfer is not a spontaneous process. Now, if we take the situation where the cathode is originally free of Na ions, does that mean the battery can be considered fully charged at the time of assembly despite not being electrically charged? 

Yes, can be considered as charged and after fabrication you need to discharge first.