Map from Google - Yakushima

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

As mentioned, precipitation is very abundant: on the coast, almost 4,500 millimetres (177 inches) of rain per year fall (4 metres, or 14 feet!), of which 770 mm (30 inches) in June, the wettest month.
In the interior of the island, on the slopes of Mount Miyanoura-dake, 1,936 metres (6,351 feet) high and covered by a famous forest, visited by many tourists and protected by UNESCO, the situation is even worse, and precipitation reaches 10 metres (33 feet): so this is one of the rainiest places in the world. The forest hosts ancient cedars, the most famous of which is called Jomon Sugi, is 25 metres (82 feet) high and is estimated at least 2,000 years old. The tree is located at 1,300 metres (4,300 feet) above sea level.
In winter the mountain can be covered with snow; on the top there are on average 11.5 °C (20.5 °F) less (the difference compared to the plains becomes even greater during outbreaks of cold air).
That's the average precipitation.
Average precipitation - Yakushima
YakushimaJanFebMarAprMayJunJulAugSepOctNovDecYear
YakushimaJFMAMJJASONDY
Prec. (mm)2752854304204407753102704053003052654480
Prec. (in)10.811.216.916.517.330.512.210.615.911.81210.4176.4
Days171518141418121514121315179

Forest in Mount Miyanoura

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

The amount of sunshine in Yakushima is scarce in winter, when cloudy skies prevail, is mediocre in spring and autumn, while it becomes 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 even 6.5/7 hours of sunshine per day.

The sea in Yakushima is cool in winter, but not impossible to try a swim; it's warm enough for swimming (with a temperature equal to or greater than 23 °C or 73 °F) from May to November, and it becomes pleasantly warm from July to September.
Sea temperature - Yakushima
YakushimaJanFebMarAprMayJunJulAugSepOctNovDec
YakushimaJFMAMJJASOND
Sea (°C)202020212425272828262422
Sea (°F)686868707577818282797572