TOKYO — Japan’s unemployment rate in April rose for the first time in six months, while the nation’s industrial production rebounded weakly from a record drop following March’s earthquake and tsunami. But output was seen picking up strongly in May and June as automakers and other manufacturers get back on line. The jobless rate edged up to 4.7 percent from 4.6 percent in March due to job losses in the retail and wholesale sectors, the Ministry of Internal Affairs and Communications said Tuesday.