You cannot ride to Cockatoo Island, and that is part of its charm.
The island was a convict prison from 1841, built to house men who had reoffended in the colony, and the prisoners themselves quarried the sandstone and dug the silos and barracks that still survive. Later it housed a reformatory for girls and then one of the busiest shipyards in the Southern Hemisphere, building and repairing naval ships through both world wars until the yard finally closed in 1991. Its industrial heritage and its convict remains are both visible, and UNESCO included it in the Australian Convict Sites World Heritage listing in 2010.
Darling Street, the main road, is lined with cafés, bookshops and old hotels, and the waterfront parks at Birchgrove and Elkington offer a good view across the harbour to the city skyline. The roads are narrow, hilly and often one-way, with parked cars on both sides, and local drivers are impatient, so allow for slow riding and watch for opening doors.
For practicalities, fuel is easy in nearby Rozelle and Leichhardt, and motorbike parking in the inner west is more generous than in the CBD, though it is usually time-limited. The ferry to the island runs from Circular Quay and Barangaroo, and a timetable is easy to find; do check it before making plans, as services vary.
If you can, go late in the afternoon, when the light on the sandstone turns gold, and watch the ferries come and go beneath the Harbour Bridge.