This is an interesting question; the answer, however, is just as rooted in Eastern vs Western culture as it is in construction techniques. Please also take into account that, as a Westerner, I can only give a partial answer based on second-hand studies.
First, take into account than on top of rot, Japanese constructions wer also subjected to earthquakes, typhoons and other weather conditions that could destroy a dwelling just as surely as mold would, and quicker. This, in general, led to a preference for structures that could be quickly rebuilt, and that upon collapse would carry a (slightly) lesser chance of causing death. I’ll return to this point later.
Now, construction-wise:
Wood is actually not as feeble a material as common wisdom would have you believe. In fact, a wooden beam is actually more resistant to fire than a steel beam, and the reason it’s not as used today has more to do with quality controls, prices and the impossibility of serialization.
Wood diseases are, in general, linked to humidity or certain types of insects, which made the problem more critical as Japan has rather stable temperatures, high humidity and a period of rains. To combat humidity in the absence of current retardants and varnishes, the solution was to provide ample ventilation, including what is known as crossed ventilation, that is, to ensure that the air flows through openings on opposite sides of the room lessening the areas with stagnant atmosphere.
Another way to ensure proper ventilation, and this one is not only traditionally but pretty much universally used everywhere, is to elevate the building from ground level. This not only allows for air to flow underneath, it also reduces the process of capillarity that would mean the building was absorbing water directly from the ground. Traditional Japanese buildings tended to reduce the weight of the dwelling, using little furniture and very light floor pieces and door mechanisms so they could further reduce the number of supports between the house and the ground; and didn’t otherwise close these open spaces.
And finally, there is the process known as micro-ventilation, which is also present in pretty much every modern window frame you see too; this is accomplished by purposefully creating gaps between the elements of the window or opening; while they are small enough to prevent rain, wind or even small objects, they still allow for further air and heat exchange.
Returning, however, to my first point, it is interesting that you asked about “withstanding” in relation to buildings, and here is where we come to the finer differences between cultures. Eastern cultures, such as Japan, weren’t as concerned with leaving a lasting legacy for the ages in the same sense Western cultures were. The idea of a building that physically outlasts its inhabitants is purely European, leading to the stone castles and cathedrals we are more familiar with; in contrast, Eastern cultures had the belief that the spirit of the building rather than its materiality was the one carrying value.
It is, basically, the paradox of the Ship of Theseus: “If you have a ship, and then you steadily repair and change pieces of it until not one element of the original exists, is it still the same ship? Even if it maintains its image and purpose?” From a traditionally Western approach to architecture, it isn’t; for an Eastern one, it is.
The result is that no element of Japanese architecture —from houses to temples or palaces— was expected to last more than a few years, two or three decades at most; it was even perfectly normal to bring them down and rebuild them as a matter of course. So the effects of mold would then, well, reset on a completely healthy wood structure.
For more information, I'd recommend you search Architectural or Art journals. Frank Lloyd Wright, for instance, apparently had a deep fascination with Eastern methods of construction. Otherwise there is:
https://www.architectmagazine.com/technology/the-history-of-wood-and-craft-in-japanese-design_o
https://www.nippon.com/en/views/b02314/
Schweitzer, Roland: "Traditional Japanese wood construction" in Internationales Holzbau-Forum 2004
Tanizaki Junichiro: In Praise of Shadows