Japanese people don't get excited about the number 13 the way Westerners do. Lacking the cultural history of the Knights Templar on Friday 13, they aren't in a position to have any particular negative associations with this number.
Instead, they are easily spooked by four and nine.
The number 4 in Japanese is pronounced ‘shi’ and 'yotsu'. The first pronunciation, the on-yomi, is a homophone for death. The characters are different, of course: 四 for 4 and 死 for death, but the sound is the same, and that’s the point.
In Japan, hospitals tend not to have ward 4. Further, products tend not to be sold in packs of four in Japan, and there’s a famous, and possibly apocryphal, story of an American golf ball manufacturer attempting to break into the Japanese market but failing miserably because their balls came in four-packs. Everything from chopsticks to dining sets is sold in packs of three or five, even though the typical Japanese family does have four members because of a tendency to stop at two children.
The number 9 is pronounced ‘ku’ (also ‘kyu’), and ‘ku’ also means suffering. The characters here are 九 for “9” and 苦 for “suffering”. The pronunciation creates an undesirable association, and so hospitals also avoid having a ward 9.
But a typical high school will have more than nine classes per grade, the the number nine is used when referring to that particular group. So despite the association, the number nine is still used frequently, at least outside of situations that are strongly associated with suffering.
None of this is much of a problem when learning the language. Japanese people generally do not expect foreigners to know their culture well, and are often surprised and even a bit put off when they do.
Instead, knowing about these numbers will help you navigate in Japan a little more carefully and precisely, since you'll know what to expect and what not to expect. You can also avoid a bit of confusion, for instance when shopping for household items. Looking for packs of four items will likely be fruitless, and asking for the ninth ward in a hospital will get you nowhere, since it probably does not exist.