Did Japan occupy Siberia during World War 2?

No, but they fought a serious battle with the Soviets (Russians) there in 1939 at NOMONHAN. This battle convinced the Japanese Army to up-grade their tanks. It also trained Soviet General Zukhov how to defeat the Germans when he transferred to the German front.

No, the Japanese Russian front was relatively quiet until the end of the war when Stalin realized he could get some territory. In the last month of the war he invaded. The Americans made a deal with him basically giving Stalin territory to the 38th parallel in Korea, thus forming a north and South Korea.