Russia might hold it for a while since if the US doesn't buy Alaska, the British are the most obvious other choice. Spain might buy it if they can somehow avert the chaos of the 1868 Glorious Revolution, but perhaps the circumstances could fall in line that Spain buys Alaska from Russia (and no doubt becomes an issue in Spain which led to said revolution). You'd have to modify 1850s/early 1860s Spanish politics to make a leader like Leopoldo O'Donnell or Ramón Narváez want to buy Alaska.
I don't think Japan is likely to buy Alaska in the 1860s-1880s period since they don't have the money and would rather gain control of Sakhalin/Karafuto and the Kurils/Chishima from Russia than Alaska. But if Spain buys Alaska, then they might pawn it to Japan in the 1880s/1890s which will be huge. Japan can easily supply the labour needed for Alaska's mines, and with their alliance with the British can fend off any animosity from the Americans. They'll settle Alaska in sizable numbers (compare the number of Japanese in the South Pacific Mandate) and before the Russo-Japanese War Alaska will be a land of opportunity if Hokkaido isn't enough. It will likely have a sizable white minority, which Japan will be careful not to offend out of fear of ruining diplomatic relations with the British and Americans.
Alaska will get Japan much more interested in the North Pacific and Anchorage (or whatever site on the Knik Arm or in the Matanuska-Susitna Valley becomes the main center). They'll likely be much more aggressive toward Russia in terms of cessions, and it's likely that in their intervention in the Russian Civil War, they'll ensure all Karafuto falls under Japanese control although they'll likely be unable to support any anti-USSR faction from ruling the Russian Far East.
Alaska will likely have some oil finds in the 1930s, but the biggest site at Prudhoe Bay will likely either be undiscovered or be impossible for Japan to tap (in any case, Japan may have more oil from Karafuto than OTL). Alaska does produce other needed resources (i.e. copper, coal) which the IJA/IJN will demand so along with the food produced in Alaska (likely farmers of buckwheat and potato--expect shochu made from those ingredients to become a major export) there may be a larger Japanese merchant fleet which will be important for them. A lot depends on 1930s Japanese politics--will they still go for expansion in China? It's possible Japan will be more cautious about offending the British or United States since Alaska with its resources is vulnerable to the Royal Navy or US Navy. I'd expect TTL's Canada will have a stronger fleet in the Pacific and measures in BC and Yukon to protect against the Japanese, not to mention the American preparation for the threat Japan poses. If something like the OTL Pacific War occurs, then it's likely Japan will be better prepared for it (i.e. larger merchant fleet), have more resources (coal, oil, and other resources from North Sakhalin and Alaska even if its just a trickle of what Karafuto and especially Alaska potentially has), and have some nice fortifications which makes the Anglo-American-Canadian advance into Alaska an extreme challenge.
It's very possible that even Karafuto and Alaska won't win Japan the Pacific War, and even though TTL's Japan will still be allowed to keep Karafuto (since the Soviets won't be able to occupy it, even though they'll likely grab most of the Kurils as compensation), Alaska will be occupied territory. It will likely be spun off as an independent republic and permanently separated from Japan. British, Canadian, and especially American interests will dominate Alaskan politics and they'll be ensuring anything remotely socialist will never win elections there. You may very well have various South Korean-style military dictatorships there which are supported in the name of "fighting communism" or otherwise have a dominant-party system like Cold War Japan. Perhaps they'll respect environmentalism less than OTL--with a charismatic leader, Rampart Dam may get built with Japanese, Canadian, and American assistance. More intensive mining than OTL will also occur and environmentalism may take a backseat to the practical concerns of the Alaskan government. Additional labour for these mines and megaprojects will likely be sourced from Japan or even South Korea.
In the 21st century, it's likely Alaska is more populated and economically powerful than OTL but at the cost of far more pollution and environmental devastation than OTL. The indigenous people may have been treated even worse than OTL. On the positive side, Alaskan culture likely has unique takes on Japanese cuisine including soba/potato-heavy dishes and famous soba/potato shochu spirits which they'll export to Japan and eventually to Canada/the United States.
Now that, that's one of the most interesting ideas, Belgian Alaska, rather than buying the Congo, if Leopold tried to make the native Alaskans mine for gold like the way the Congolese were abused I wonder if that could spark British intervention, maybe they just outright snag it for that.
There were nowhere near enough of them to force into mining gold and the Tlingit were your archetypical "indigenous group pulls off an intense resistance against modern forces" sort of people. The Aleuts might be easier to manipulate if the Belgians can inherit the state of subjugation the Russians placed on them.