You are on a cruise ship - there is no need to stay in a hotel - you sleep on the ship.
You should hopefully have a better idea of where you are going than we do, but if you refer to an atlas you will find that Spitsbergen is part of the Svalbard group of islands way north of the North Cape. So if your cruise goes there then it's an option but if you only go as far up as Honnigsvag you are still hundreds of miles away.
But before you get too involved just make sure there is an ice hotel there. The famous one is in Sweden but even that is only open December to April and melts in the summer.
These are your ports of call - it does not mention Spitzbergen - Amsterdam, Holland, Alesund, Norway, Arctic Circle (Cruise), Alta, Norway, Honningsvag, Norway, Tromso, Norway, Molde, Norway, Olden, Norway, Bergen, Norway, Amsterdam, Holland.
Spitzbergen is a protected port as is everything on it - wildlife, tundra etc. There are no hotels here of any kind, let alone an ice hotel; nor are there any cafes, bars etc. When walking around you have to keep to the marked paths. However, it is a stunning place if you like nature because that is all there is - nature at it's rawest but with a beauty all of it's own. I loved it! Regards - Jane.
My wife and I did this trip last year on the Century. It was one of the best cruises that we've been on. The full day trip in Alta including the husky farm visit was great. Cant recommend this cruise highly enough.