The annoying thing is that there is no direct link, either by bus or railroad, between Kandalaksha and Zapolarny. You’ll have to travel through Murmansk. But it turned out easier than anticipated.
To get to Murmansk fast and cheap hop on a minibus that leaves at 5:30 am and 6:30 am off the train station parking lot. To reserve a seat call 8 911 349 9000. The trip will take about 4 hours and as of the moment, it costs 700 roubles ($12US). The train is slightly slower and considerably more expensive ($20-35) but there are several during the day. See tutu.ru for the train schedule.
Another option of getting to Murmansk is by arranging a ride through blablacar.ru. My first experience with the system was highly positive. Lots of traffic between Kandalaksha and Murmansk, and the “standard” cost is 500 roubles (under $10) per passenger.