The cheapest transportation option in Honolulu is The Bus. The Bus has received several awards over time by the U.S. Department of Transportation because of its buses' punctuality. Each one-way fare costs $2.50. Upon boarding, you can request the driver a transfer that will be valid for 2 hours and allows you to make one free transfer. Children under 5 years old travel for free when accompanied by an adult. From the Honolulu Port, it's easy to walk towards Chinatown, where you will find that several bus lines stop on the corner of Hotel Street and Bethel Street.
For more info, see our feature Honolulu Port FAQ