What if you don't have a car or your car break down, are you allowed to transport using public/private transportation?
The Bus is an interesting question. Revised Ordinances of Honolulu, Chapter 13, covers The Bus. Nothing in its section on prohibited activities mentions firearms, although there is a prohibition against:
Carrying or possessing any flammable, combustible, explosive, corrosive, or highly toxic liquid or other substance, article or material which is likely to cause harm to others or to emit any foul or noxious dust, mist, fume, gas, vapor, or odor; provided that nothing contained in this subdivision shall be construed as prohibiting a person from carrying or possessing any match or any cigar, cigarette, or pipe lighter, which is not lighted or smoldering;
So I called their customer service number. Talked to a very helpful person. Turns out that "guns and weapons" are prohibited under what she called the Modified baggage Rule, city ordinance 4635. I've been unable to locate that particular ordinance online, but she had it right at her fingertips.
So, if you break down, call a friend. If you can't find a friend, call a cab.
I did break down once on the way to the range. I called a tow truck, and had him swing by my house on the way to the repair station. Cost me a few $$$ in extra mileage, but the guns never left the 'security' of my car, and never left the most direct route between my house and Kokohead.
I lived without a car for several years, only owned my motorcycle. With saddle bags and bungee cords, I could manage a range trip pretty well. I built a PVC over-the-shoulder case for longarms, which I eventually replaced with a partial golf club bag. Did that every weekend for, oh, 3 years. Had HPD behind me from time to time, as happens in the normal course of traffic. No one ever pulled me over or looked at me funny.
HRS 134, to the best of my recollection, doesn't address HOW you transport it. Pushing it in a shopping cart down the shoulder of the road may be every bit as legal as having it in your car.