How much air does a 777-300ER's pressure cabin hold?
I once read it's about 1 metric ton (1000 kg) of air for a large widebody.

Here's a ballpark calculation to verify this.

The 777's cabin altitude is 2430 meter (8000 ft) when it's cruising at max altitude.

Wolfram Alpha tells us that a cubic meter of air weighs 0.96 kg at that pressure.

DimensionInfo shows that the 300ER is 73.9 m long and 5.86 m wide.

For simplicity, consider the 777 to be a perfect hollow cylinder.  The volume of the cylinder is:

$$\pi \cdot \left(\frac{5.86}{2}\right) ^ 2 \cdot 73.9 = 1993\: m^3 $$ 

So that's about 1900 kg of air for the cylinder.
You have to subtract a large amount for the air lost to non-pressurized volume, tapering of the cylinder, non-circularity of the fuselage, and volume not occupied by air (structures, furniture, insulation, wiring, luggage, passengers).

I'd guess it's somewhere between 1000 and 1500 kg.

Whether people are on board or not has very little influence.  People float or sink in water depending on having air in the lungs or not.  So the density of a human is around that of water: 1000 kg per cubic meter.  [In other words: you could fit 10 people weighing 100 kg in a cubic meter if you pressed them hard enough].  So the density of air is a thousand times lower than that of the passengers.  Three hundred people - weighing 30000 kg combined - displace less than 30 kg of air.