look, I don't have any potentiometer lying around, and I don't have any variable resistor. So is it possible to instead use a fixed resistor for the LCD display ?
Sure. You can use two resistors in series to be a "fixed value" potentiometer.
The "wiper" of the potentiometer is the junction between the two resistors.
I can't tell you what values to use, so you'll have to experiment. You want them to add up to somewhere in the tens of kilo-ohms though, ideally, though the value isn't that critical - it's the ratio between the two values that is of interest, and it's that which gives you your contrast setting.