Kirkenes Norway - after the ferry cruise

Have you looked into flights? I can’t help but think that would be quickest/easiest option.

Heads up: far above polar circle there are no trains in the far north. Winter and so. Not enough people and so.

Idea: fly to Bergen - NiN to Oslo - fly / bus to Stockholm - Tallink to Helsinki.

Or spend time around Kikenes or Tromso for chasing Northern Lights - if not seen on the ship.

Not doing the round trip means that you are missing half the scenery. The timing of the ship is such that what you pass by night on the way north you pass by day on the way south and vice versa.

Apart from the ship there are only two ways out of Kirkenes
Fly-at least as far south as Narvik.

Bus to Rovaniemi in Finland for the Arctic direct day or overnight trains to Helsinki.

That is the daily bus at 0616 to Karasjok, arrive at 1110, overnight there, then next day bus at 0915 arrive Rovaniemi 1725. Either stay overnight in Rovaniemi or that is a nice connection into the overnight train.
This is the bus from Karasjok- https://www.eskelisen.fi/en/product/ticket/#matkakauppa-search-results
and this the Kirkenes to Karasjok bus- https://snelandia.no/

Or- Overnight train from Narvik to Stockholm, then overnight cruise ferry to Helsinki or Tampere (for train to Helsinki- but note that this line currently has long term bus replacements due to engineering work).

Two companies run the ferry from Stockholm to Finland- Viking and Tallink Silja

The challenge is getting out of Kirkenes, and how much of the arctic you want to see. If the answer to the latter is as much as you can then the bus is the best option.

This is the Finland Railways official website- https://www.vr.fi/en
This is the Vy Norrtag website for Narvik to Lulea and Haparanda- https://www.norrtag.se/

If you fly to Narvik then there is the train to Lulea (overnight stay) for the international train to Tornio/Haparanda (Finland) then day and night trains direct to Helsinki. The twin towns are Haparanda on the Swedish side and Tornio (over the bridge) on the Finland side.

Note that there is normally a better train service from Narvik (including an overnight train to Stockholm)- currently it is constrained by track repairs after a freight train derailment- the time frame to complete repairs is a bit uncertain.

Thank you all for good advice. Especially @isn31c it was very clearly defined and I appreciate your specific bus routes and stopovers. That said, I relooked at the Hurtigruten cruise round trip and decided it would be OK to do that. It makes sense with our itinerary.

Betsy