So you've been in Sydney for a while, safe and sound hopefully.
Christmas can be a bit expensive as far as booking in anywhere and have you got a hostel I imagine and if they aren't doing anything, you may find a few about like yourself at a loose end and you could see if you can drum up a bit of spirit, pool resources and organise a feast or if just the two of you, what about getting a hamper of goodies and heading to the beach somewhere for a picnic - you may even find some families that extend Xmas greetings into an invite to join in..
You'll probably find just about every backpacker in Sydney thinking of the same and Bondi is usually where they head [or Manly], and heard on news that the local council has a problem with some homeless taking up residence in the Bondi Pavillion!
Talking of homeless or older people, if you're feeling charitable at all and to do something different I've heard in past of people helping out at homeless shelters or ol peoples homes where they usually attempt to put on something special but of course, they'll likely have less workers as some will take time off to be with family.
Ring up the councils if you feel so inclined and you could find some places.
If not, seeing as Bondi and Manly will be extra crowded, think of trying some other beaches, the South Head being a nice area and good for sea breezes if it's warm, and a few nearby harbour beaches or head up to the northern beaches, Long Reef headland just before Collaroy quite nice or right to the end at Palm Beach and some buses should be running from the city - 131500 Transport Infoline - Home
Also get yourself out to South Head on Boxing Day morning to see the start of the Sydney - Hobart Yacht race as that is quite a spectacular sight of all the yachts and pleasure boats streaming down the harbour.
And for NYE, it'll be hard to miss the droves of people from before midday all heading towards the harbour to stake their spots for the Bridge fireworks and the Harbour area gets packed.
A good trick to be less packed may be to take a ferry over to Milsons Point or walk across the Bridge and then down to around where there are some small parks/ferry terminals as you'll likely have less people.
Again, a picnic hamper of sorts may be the go.
And beforehand, just in getting about the harbour area, check out what pubs might be open on NYE and then you'll have an idea of where to try and get in for a drink afterwards, but expect most to be overflowing too.