Ryukyu Islands, map

The climate of the island of Yakushima is humid subtropical, with mild winters and hot, muggy summers, and it's famous in Japan for being rain-soaked all year round.
The island is located at the 30th parallel, south of Kyushu, and together with other islands such as Tanegashima, Kuchinoerabu-jima, Kuroshima, and Iojima, is part of the Osumi islands, the northernmost of the Ryukyus. In winter, it is affected by weather disturbances due to the clash between cold air masses coming from the Asian continent and mild and moist air masses that lie on the East China Sea. In summer, there are periods of bad weather due to the summer monsoon, as well as to typhoons.
The temperatures are similar to those of Mediterranean islands such as Sicily. In winter, the temperature never drops below freezing, but there may be cold periods, during which highs drop below 10 °C (50 °F), the wind blows, and lows approach 0°C (32 °F). In summer, hot and sticky weather prevails, although sea breezes blow.
Here are the average temperatures.
Yakushima - Average temperatures
Min (°C)9911141721242423191511
Max (°C)141518212427303129252117
Min (°F)484852576370757573665952
Max (°F)575964707581868884777063

As mentioned, precipitation is very abundant: on the coast, more than 4 meters (14 feet) of rain fall per year, of which 770 millimeters (30 inches) fall in June, the wettest month. Here is the average precipitation.
Yakushima - Average precipitation

In the interior of the island, on the slopes of Mount Miyanoura-dake, 1,936 meters (6,351 feet) high, the situation is even worse, since precipitation reaches 10 meters (33 feet): so, this is one of the rainiest places in the world. On the mountain, a famous forest is found, visited by many tourists and protected by UNESCO; it hosts ancient cedars, the most famous of which is called Jomon Sugi, which is 25 meters (82 feet) high and is thought to be at least 2,000 years old. The tree is located at 1,300 meters (4,300 feet) above sea level.
In winter, the mountain can be covered with snow. On the top, the temperature is on average 11.5 degrees Celsius (20.5 °F) lower than in the plain (and the difference becomes even greater during outbreaks of cold air).

Forest in Mount Miyanoura

As mentioned, in summer and early autumn, from June to October (and more likely from August to early October), Yakushima can be affected by typhoons, the tropical cyclones of Southeast Asia.

The amount of sunshine in Yakushima is scarce in winter, when cloudy skies prevail, it is mediocre in spring and autumn, while it is decent in July and August, at least on the coast, when the rains are still abundant, but they are not so frequent, and there are 6.5/7 hours of sunshine per day. Here are the average sunshine hours per day.
Yakushima - Sunshine

The sea in Yakushima is cool in winter, but not impossible for a swim. It's warm enough for swimming, with a temperature equal to or higher than 23 °C (73 °F) from May to November, and it becomes pleasantly warm from July to September. Here are the average sea temperatures.
Yakushima - Sea temperature
Temp (°C)202020212425272828262422
Temp (°F)686868707577818282797572