You must check weather forecast before go to mountains.
First, check whether it will be bad weather or not. If it will be bad weather, call off or change the plan.
Second, check the temperature. Most weather forecast sites show only temperatures in the major cities (often in lowlands). So you must estimate the temperature in the mountain using the law that the temperature drops by 0.6 °C per 100 m. You should select the suitable clothes and equipment based on the estimated temperature.
In the mountains, weather tends to be bad in the afternoon. You had better start climbing or trekking early in the morning and finish at 3 pm at the latest.
Weather Forecast Sites
As for the weather in Japan, forecasts by Japanese institutions are most accurate. This page introduce weather forecast sites by Japanese institutions.
Japan National Tourism Organization
Simple and multilingual 10-day forecast.
Japan Meteorological Agency
Weather forecast, weather chart, various data and information of weather, earthquake and volcanoes.
Too much information, difficult to read.
tenki.jp (Japan Weather Association)
Japanese only. Most recommended Japanese weather forecast site. Abundant weather information and commentary.