When travelling, it's generally smart to pack in preparation for rain or any other conditions that may arise. The rainfall in Doha is roughly less than Pusan. You can assume this to be the case over the next 1 days.
Expect the weather to be around 42 °C on average in June. As the temperature peaks, try to get your plans done early, as flight price and availability will have an impact on your travels. The climate in Pusan is generally colder than in Doha.