Hammerfest does indeed have an airport (Airport Code: HFT). It is about a 10 minute drive from the town center. It is served by Widerøe Airlines, which provides multiple daily flights between Tromsø (TOS) and Kirkenes (KKN) stopping at several arctic villages along the way. There is no bus service to or from the airport, though taxi service is possible.
Northbound coastal steamers arrive daily at 5:15AM, departing for Kirkenes at 6:45AM. Southbound steamers arrive daily at 11:15AM, departing for the journey to Bergen around 12:45PM.
To get to or from Hammerfest, you'll normally take a "port-to-port" trip. The website has a list of up-to-date fares and cabin options. Online booking is possible, but for those living outside Scandinavia, your best bet is to email Hurtigruten directly if you want a cabin. Book well in advance if you're traveling in June or July.
Also check the website for the latest specials.
I use to fly there with the local Wideroe
They have good connections from Tromso. The price may be a problem, but if you
really want to go there this is the best solution I believe.
Another idea is to use the Hurtigruten: http://hurtigruten.no
I hope that this is what you are looking for.
If not and if your time is not limited... you can try to go there by car, motorcycle or, why not, by bike.
If you need any other details... I am ready to help!
Like always, it might be the fastest way to go by plain to Hammerfest (there are severaly flights from Oslo), but definitly the most beautiful way is by train/bus. Trains are running in Norway until Bodo - or if you come from Sweden until Narvik. From there you can take the coach up North, a really beautiful trip in a comfortable and modern coach. You'll pass all the fjords, high cliffs, steep gorges and an unique beauty of nature: Norway in your wildest dreams!
http://www.nor-way.no/tagbusstysk/zugbusty.htm A good Site for Busses in Northern Norway.
All the way up north there are usually good signs of directions how to you’re your way, however, once during a severe storm I did miss the right turn and found myself in another place ;-)
There are some nice bridges on the way to Hammerfest which saving you lots of driving distances if you would need to drive around the fjords.
I am not sure about public transportation, because of the mountains, there are no trains here, we used our car. You can also fly, I am not sure if there is airport here but there is one in Tromso.