Kirkenes is at the very end of the all-through-Norway (and half of Europe) highway E6. You can also enter by road from northern Finland (via Rovaniemi about 7 hours), and northwestern Russia (via Murmansk about 3 hours + border formalities). If you want to come from southern Norway by car, cutting through Sweden and Finland is actually faster. Long distance buses , with connections to most of Norway. Buses to Finland (summer only): 
Kirkenes is the northern turnaround port (Bergen being the southern) of the Coastal Express , arguably the most beautiful sea voyage of the planet, covering most of the Norwegian coast in 5-6 days.
Due to its distance from the rest of the civilized world, most people choose to arrive and depart by plane, at Høybuktmoen airport (IATA: KKN) some 15 km outside town. Daily flights to Oslo, Tromsø and others by Norwegian , SAS and Wideroe . Also flights to Russia and Germany (summer only). Most flights have a bus to town waiting outside. A taxi will set you back some NOK 300. On certain occasions it is considerably cheaper to fly into Ivalo in Finland, some 240 km away.
Although there is a railroad here, it is for ore only. Real adventurous spirits should be aware that you can connect to the russian network in the border town of Nikel, travelling as far as Moscow, Vladivostok, Ulan Bator and Beijing.
Local buses .
Fishing, both salt and fresh water, including salmon. Hiking.
Try Ritz or Ofelas.
There is a camping site between the airport and town, up the hill after the bridge as you drive towards town. Also rents cabins. Summer only. Tel (+47) 78 99 80 28.
If you plan on crossing the border into Russia, you will probably need a visa. In theory, this should be obtainable at the Russian Consulate in Kirkenes, but you might find it worthwhile to pay any local travel agency a little extra for the job.