Yes. All you really need are paths. This is a bit of an overgeneralization since forests in different places have different growth habits, but many old growth forests have less underbrush. During the winter, these forests are pretty easy to navigate on Nordic skis. If he wants to skate instead of doing the classic stride, I’m sure he can afford to groom the paths. Grooming really just requires a fancy snowmobile.

It's not far from a skiing resort.

https://funasdalen.se/en/winter/skiing/cross-country-skiing/

Potentially. Depending on the style of skiing, you may only need a 2-3 ft wide pathway. That said, ski paths tend to be cleared of trees/large rocks and span about 10-15 feet wide.

Who said it's a virgin old-growth forest?

It might not get to remain a virgin old-growth forest. If the Swedish logging companies can get away with it, he probably can, too.

Alpine versus cross country.

I wouldn't be surprised.

Karp strikes me as the type of person that would do this "for fun".