A taxi or the train are both okay options from the airport to central Sydney. The taxi will cost a bit more but might suit you if your hotel is not near a railway station or you just want to minimise the hassle. Your flight times are such that you might avoid peak hour on the roads - when the train can be quicker.

Yes, it's possible to take a public transport option that avoids the airport train charge, but you should not waste your time with that on your short trip.

If you use public transport at any point in your visit be aware that you may not need an Opal card - they claim the system now works with any contactless card such as a credit card. This should save you from sacrificing any unused credit if you were to buy an Opal card. [Can't post link but you will find it under 'Opal News'.]

I'm not a luxury traveller and don't stay at hotels in Sydney city but I will venture to suggest: If you like historic buildings or want to be right near the big sights of the harbour bridge and opera house you might appreciate staying in the area called The Rocks or other places at the northern end of the CBD. Prices are high and depending which street you are in you will be surrounded by other tourists or office buildings. Mid-week it can be a bit lacking in street life north of about King Street. If you like lots of street life and interesting ethnic food (mostly Asian) then the southern part of the CBD towards Central Railway might suit you. It's still less than a mile to Sydney Harbour but you won't see it out of your window (unless you are on the 30th floor!). There are lots of big charmless hotels on the western side of Darling Harbour near the casino. Not recommended unless you like charmless casinos.