Bourne Russell captained the "Lady Rowena" to Australia in 1829. On board was an apprentice sailor named William Watson. How do I verify this William was the same William born to Bourne Russell and Elizabeth Watson on 30 June 1815?
In order to build the case that these are one and the same person, you need to build a chain of links. Something like the following might suffice.
Of course, this is not proof but sources such as the Trove Newspaper database should allow you to gather evidence that is strongly suggestive of the claim.