While it would seem obvious that John Bolton would/should get some sort of clearance from CIA/DOD prior to publishing his book, lest he reveal some classified information,does Bolton have any obligation to obtain clearance from the White House to prevent him from disclosing information that might be considered covered by executive privilege?
He would almost certainly have signed a non-disclosure agreement granting the CIA and the NSA the right to review the book before publication. Edward Snowden got hit for violating that clause
A federal judge in Virginia has ruled that the U.S. government has the authority to seize the proceeds of Edward Snowden's book because he failed to submit his book to the CIA and National Security Agency (NSA) for pre-publication review to analyze any classified information printed within.
While Executive Privilege is a recognized right of the President, it simply means that White House staff may not be forced to disclose private conversations within the Executive branch. It does not mean they cannot discuss them willingly
Last year, Trump's campaign filed for arbitration against Manigault Newman, alleging she violated a nondisclosure agreement by publishing a tell-all book. Her book, "Unhinged," painted the president as a racist and an incompetent leader.
Manigault Newman, who worked both for the 2016 Trump campaign and in the Trump administration, released secret audio recordings of Trump, Kelly, and Trump associates Katrina Pierson and Lynne Patton that she says back up explosive claims in her book.
In other words, Omarosa had a non-disclosure agreement with the Trump campaign. I have never seen anyone claim she violated Executive Privilege. Since Bolton did not work for the campaign, it's unlikely he would have to be concerned about this for clearance.