I know this is an old post, so mostly documenting this in case someone besides me comes across it.
If you're greasing any FANUC robot, I suggest getting a hold of the Mechanical Unit Maintenance Manual for each robot from FANUC. Sometimes you have to have some joints at a certain angle when replacing the grease.
Long story short, you need to remove the grease "Exit" bolt and then you can start pumping (very slowly) NEW VIGO RE0 or MOLYWHITE RE00 grease into the joint. Pump until clean, new grease is coming out (or there is very little gray/black crud in the grease coming out). I would personally rather was a little new grease than leave contaminants and crud in the grease joint.
After the grease has been added, you should jog the robot around, as specified in the manual. Essentially you need to jog enough to get it to "burp" any air bubbles out of the grease and make sure the grease pocket is full of new, clean grease.
On bigger robots, like R2000 and M900's, you will use VIGO RE0 (looks like Honey) for all joints, but the balancer will be greased with a different grease, I believe it's any grade 2 grease.