Murmansk is on the Kola highway(M18 or E105) from St. Petersburg to Norway. Comfortable travel time by car is 4-5 days from Moscow or 3-4 from St. Petersburg but if you put your mind to it you’ll make it in two.
Highway P12 will take you to Murmansk from Finland.
Airport “Murmansk” is actually in Murmashi, 24km from Murmansk. Daily flights to Moscow and St. Petersburg. Regular flights to Arkhangelsk, Tromso (Norway), Helsinki (Finland), Turkey, and Egypt.
3-4 direct Moscow to Murmansk trains, and several more from the South of Russia passing through Moscow on their way to Murmansk daily. Total time in transit about 40 hours. Second class tickets from Moscow to Murmansk are from $80 to an equivalent of $170 as of April and May of 2013.
Most Moscow to Murmansk trains go though St. Petersburg, some via Volkhovstroy. If you have a choice and don’t need St. Petersburg take the Volkhvostroy one. These are much less crowded, and in winter downright empty. No need to buy tickets in advance except during travel season and holidays.
Travel time from Kandalaksha to Murmansk is about 6 hours. A Russian driver will make it in 3-4. Lots of private shuttle buses going back and forth. Details soon but once in Kandalaksha you’ll have no difficulty finding a ride.