Alexander appeared in 36 episodes of the New Zealand soap as Marc Peders.
In a video post on social media, he said he started his hunger strike after a rally in Christchurch over the weekend in support of Palestine.
He vowed to continue until the government met a list of demands.
"I am now in my second day of this zero calorie hunger strike and have no intention of stopping until my demands are met," he said.
His demands include "the withdrawal of New Zealand troops from the Red Sea and the resumption and then doubling of the humanitarian funding for UNRWA, the United Nations Relief and Works Agency".
He also called on "NZ company Rakon to stop supplying components for weapons used by Israel".
"If these demands are not met, I may soon be dead like thousands in Palestine and Israel," he said.
The Government deployed New Zealand Defence Force personnel earlier this year to "uphold maritime security" in the Red Sea after attacks by Houthi rebels on commercial and naval shipping, the New Zealand Herald reported.
Said Alexander: "It doesn’t matter whether our troops are cleaning latrines or pressing the button. Their presence there means that New Zealand is actively supporting the bombing of Yemen just to keep weapons flowing into Israel."
At the time of the deployment, the Herald reported Foreign Minister Winston Peters said the international coalition had made strikes against Houthi military targets which had played a role in attacking commercial and naval vessels.
"These efforts support international security and the free flow of trade on which New Zealanders rely," Peters said.
