Get the lanyard for your room card and keep it around your neck. It makes getting on and off the ship a lot easier when you are all bundled up.
In Alesund you get the whole day in winter as they can't go in the fjord. Book a taxi (there's a Norwegian taxi app) to pick you up from the terminal and spend at least half the day at the Atlantic Sea Park -
https://en.atlanterhavsparken.no/
Do the Tromso tour with the Arctic Cathedral and Cable car.
In Kirkenes, be ready as soon as the ship docks. There's a little wooden cabin on the right as you get off, they sell the most fabulous Russian dolls and souvenirs. Just after there, hang a right through the industrial estate, it's a short cut into the town.
Be prepared to do a lot of walking. Sturdy waterproof boots, with thermal boots and yak traks will be fine. Practice with the yak traks as you need to remove them when you go into buildings. You only need one other pair of shoes for onboard, I just take Uggs.
There's a packing list on the Hurtigruten website. Get thermals from Landsend. Have a windproof coat and trousers. Take woolly hats including one for the hot tub (also sliders, the deck gets icy and your feet will thank you). It can be a bit nippy in the bar by the big windows, take a small fleecy blanket for your legs. Go to the lectures and the on deck stuff. Get the sheet on camera settings. If you can, take a small tripod for your camera, it's much easier for photographing the Northern Lights. Take a book or two.
I've done it a few times now, it's a really enjoyable and calming holiday. Life on board has a rythym, enjoy it.