You have 2 options. The first one is to take a cab, which is not only the fastest but is also the most efficient way to get to the Cruise Port. You can expect to pay between $15 and $20 for a 10 to 15 minute drive. The second option is to take the bus. You will need to pay a fare of $2.50 per adult. Bus line 19 in direction of "Waikiki Beach & Hotels" stops at the Cruise Port. Do not take the bus that heads towards the Hickam Naval Base. Only take the bus if you are not in a rush.
For more info, see our feature Honolulu Port FAQ