If Obama is “ineligible to be President,” then I don’t think he’s eligible to be “acting President,” either.
However (I heard “here we go again” - I’ll have to dig up my past posts on this subject - here’s one from 2012), technically, Obama is not “ineligible to be President.” He is ineligible to be elected President. I don’t see anything in the Constitution that says that he cannot become President through other means.
(22nd Amendment, Section 1, first sentence: “No person shall be elected to the office of the President more than twice.”)