When service to a single station, as opposed to an entire line, is discontinued as in the case of Kami-Shirataki station, the trains which used to stop there are still running, they just no longer make that stop. So it's possible to "pass by" the station by simply taking those trains.
Whether it is easy to actually go to the station obviously varies, but your best bet is probably to take a train to the nearest (still operating) station on the line, and walk to the defunct station, as it is unlikely that bus service to that station will be maintained. If the station is near a population center, however, there may be some bus service to a nearby place... Every case is different. In the case of Kami-Shirataki station, it is "only" a couple of kilometers from Shirataki station, which has much more frequent service (it is even served by Okhotsk limited expresses).
Of course, if the entire line is being discontinued, buses are most likely the only public transport option (if they exist), but it is also possible that there is another line which has a station nearby (that would be more likely in urban settings).
Usually, the building is left, but here again each case is different, it may be destroyed (if there was a building at all...) if it poses some sort of risk.