On the video, which shows a static image of Goto holding the photograph, a man's voice claiming to Goto's says Japanese Prime Minister Shinzo Abe is to blame for Yukawa's death.
The voice says his captors no longer want money, but rather are demanding the release of Sajedah Rishawi, who is being held by Jordanian authorities.
"You were given a deadline, so my captives acted upon their words," he says.
"They no longer want money, so you don't need to worry about funding terrorists," the voice says in English. "... Again, I would like to stress how easy it is to save my life. You bring them their sister form the Jordanian regime, and I will be released immediately -- me for her."
CNN's Ali Younes and Yoko Wakatsuki contributed to this report.