Upvote:5
There is some duplication with the techniques mentioned in this question. However, if Matthew is travelling a lot in that area, chances are he travels with the same airline a lot. So probably his best option is to make sure he always travels with the same airline or alliance and uses their frequent-flyer programme. Sooner or later he'll get upgraded to an elite level and start getting lounge entry for free (as well as other benefits such as faster check-in, security, and other good stuff).
It's probably best for him to pick an airline/alliance based in his hometown, assuming that's where he always flies from/to.
One good option (not the only one) in that area might be Finnair (part of the Oneworld alliance), who have their hub in Helsinki.
Upvote:5
One option is to use a PriorityPass which has 2 matching options: 250 euros every year for 10 free lounge accesses and then 24 euros each, or unlimited access for 400 euros every year. They offer lounges in Helsinki and Talinn, with some conditions on the time of the day or destination. But there is no lounge in Oslo!
So another option is, like @Andrew said, to become a frequent traveller on a specific airline. Scandinavian Airlines has lounges in Oslo and Helsinki and has partnership with IcelandAir, Estonian Air and Air Baltic when buying a business class ticket on these airlines (and they're member of Star Alliance). Talinn Estonian Air lounge is accessible to EuroBonus Gold card holders (SAS frequent flyer program) so I suppose this program would be a good choice. I don't know how much you need to fly get this status though (it sounds like a lot of points but maybe it's easy to get them)