I want to use Electronic Fluorescent ballast for some conversion of a luminaire i am doing

The luminaire is made out of a "main part", a stick on which it is suspended from teh ceiling, and a small "cup" for connections at the top of the stick, near the ceiling. It is all metal, and it is Earthed by an Earth wire to the "stick" at the top (in the connection box)

I want to place the ballast in the "main part" of the luminaire

The ballast is with Plastic enclosure. It does have Earth terminal though. Inside, this Earth terminal goes to one Y capacitor, the other end of which is on the Negative side of the internal DC line (the Electrolytic capacitor). The ballast have PFC, where the Negative of the Electrolitic comes straight from the bridge rectifier, the Positive through the PFC circuitry

Is the Earth connection for this ballast critical for something ? Or i can skip on pulling the Earth wire specially for it through the "stick" ?

Yes, it is critical for the EME, it uses to be way more efficient than all the filter chokes or so. It has to be connected by as short as possible lead to the reflector behind the lamp, as that is, where the HF current come from.
It forms a current return path for the current "leaking" through the capacitance between the lamp and reflector.
Without this capacitor, the current will flow through the mains input wiring, so then get radiated (and so cause distturbance) by the house wiring.