OP specifically asks for a culturally russian Alaska. This would create a japanese Alaska, which while an interesting scenario, doesn't fit with the thread.
I think the Whites fleeing to Alaska is the best bet, but I'm not sure how many you could actually get there, considering how far Alaska is from just about every significantly populated part of Russia (the closest would probably be Vladivostok) and if I recall right, weren't most of the Whites on the opposite side of Russia from Alaska?
He didn't say it had to survive into the present day... or that it couldn't be occupied at some points in it's history. The Russo-Japanese war is just my explanation for how it is separated from Russia proper.
Also I never said Japan holds on to it till present day, (as that was not the case with Sakhalin) or that the Russian whalers, miners, hunters etc. living there were removed by the Japanese.
OP compared his hypothetical Alaska to Mexico, which was also occupied by a foreign country for a while (France), but is still culturally similar to Spain.
I just meant that post to be a jumping off point for other ideas, not an end all be all timeline where nothing happens after '37.
As far as the civil war goes, Allied forces were holding the Trans-Siberian Railway for quite a while. A fraction of displaced whites could certainly take the train to Vladivostok and from there get a boat to Sitka (or Novo-Arkhangelsk, as it'd likely be called)
Here's a map of the war: