As others have said Darling Harbour is a good place to stay, I would think an Aparthotel would meet your needs best. Essexexile mentioned Pyrmont, you would probably find Airbnbs around there. Really when staying in Sydney city you are choosing between Darling Harbour/Chinatown, The Rocks/Circular Quay or the hotels on George St./Pitt Street, with your age range I would think Darling Harbour best as you are more likely to find Aparthotels there and it is super close to Wynyard train station, It is a beautiful walk along the water past lots of bars and restaurants through Barangaroo, past Walsh Bay to The Rocks. Sydney dance company and Sydney theatre company are based at Walsh Bay and tickets are definitely cheaper than London prices so maybe a show? The Opera on the Harbour is usually on around Easter time too, if not the Opera House is worth a visit and you can have drinks with great views at The Opera Bar.
Other things I would recommend:
Walk across the bridge to Kirabilli and explore (there is a monthly market under the bridge)
Take a ferry (not a harbour cruise) either to Manly which is lively, lots of restaurants and bars and nice beaches or to Walsh Bay for fish and chips (look longingly at the waterfront houses on the way)
Ghost tour at Q station (old Quarantine station where the ships from Europe docked), or just a wander and lunch there
Cookatoo Island is interesting
Take the B line bus up to Palm Beach, get off at Newport for The Newport pub which has gorgeous views over Pittwater or continue on if you are interested to see where Home and Away is shot
Hire a boat at Palm Beach and take it over to The Basin or one of the little beaches only accessible by water with a picnic (or there is a ferry)
Eat/drink at The Glenmore rooftop in The Rocks (shame the cruise ships will be back soon spoiling the view!)
Carriageworks - art and markets
Botanic Gardens and Art Gallery of NSW in The Domain
Lots of little underground small bars in the CBD, google as they can open and shut
Spice Alley in Kensington (near Central) is great and Koi dessert bar is there too
Paddington markets on a Saturday morning with lots of great places nearby for a pub lunch
The coastal walk from Brontë to Bondi is stunning (you could continue on from Paddington by bus)
Da Orazio pizza and porchetta has just reopened in Bondi, great for lunch
i could go on…..
Most Sydney visitors do a trip to the Hunter Valley wineries but without transport you will be relying on an organised coach trip. Choose very carefully, the small ones can be great, big coaches going only to the big wineries full of people wanting to get drunk less so! Ideally you want one visiting boutique vineyards stopping somewhere nice for lunch 😊 You can get a train from Central to Katoomba in The Blue Mountains but I think you would have to use the expensive tourist bus to get the The Three Sisters or Scenic world (which is worth a visit, spectacular views from the cable car that goes across the valley) so you might not want to give up a full day.
Have a great trip.