As eva said, you probably won't find a shuttle to the airport or cruise terminal. The cruise terminals are close to the city centre. There is a FAQ guide to getting to and from the airport.
The Rocks was the slum of the port of Sydney in the late 1700s right up to 1960s. Many of the small buildings, narrow streets and warehouses have been preserved, and now much of the area is now redeveloped and marketed as an historic precinct.
However, the grand Sydney buildings were built in the present day city centre. Macquarie Street on the eastern edge of the CBD has The Mint, The Hyde Park Barracks,, St James Church, and even St Mary's Cathedral and the Mitchell Library. Behind that the Greenway designed Government House. So, the history of Sydney isn't just found in The Rocks.
I'm sure you'll have no problem finding a "decent but nothing fancy" hotel in the city for under $200. Try wotif, and check reviews here. And it isn't far to walk to The Rocks from the city centre.