Charma j, your visiting dates for Sydney are a bit awkward, in that you're going to have to research more thoroughly to see the sights. I guess it is also important to consider whether you wish to celebrate Christmas Day with a special lunch or dinner, or whether it will be just another day in your itinerary.
I would think that most publicly-owned attractions (museums, etc) won't be open, but then many of the best things about Sydney can be seen without an entry fee. By this I mean you will still be able to see the harbour, walk around the city, view the Harbour Bridge, go to the beach or do a beach walk etc.
The shops will be open later on Christmas Eve; some convenience stores will open on Christmas Day; and the large department stores in the city will open for the Boxing Day sales.
Public transport will operate, but on a reduced timetable, so it will be important to check this out. If the times work, you could still do a ferry ride to Manly, say.
Restaurants in major hotels and in prime areas like Darling Harbour and Circular Quay (pronounced "key")will open but will probably require an advanced booking and will be expensive, if you want a special Christmas meal. If not, you will probably find a small store that sells some take away - fish and chips by the harbour on a beautiful day is not a bad thing!- or buy some staples when you arrive on 24th so that you've got the makings of a picnic.
Many attractions will reopen on 27th (probably quite a few will open on 25th and 26th). However, to make the most of your time, research now what you would most like to see and do and then check those websites for their opening times over the Christmas period. I just checked and the Bridge Climb, for instance, operates on Christmas Day.
I hope you have a wonderful stay!