Other than the part about the president being immune to such things, your comment is wrong. Other than not having standing nor an actionable claim, he could sue the agency whose contract was mandating the vaccination regardless of who put the mandate they were enforcing. Of course, you're required to follow the rules of the CDA or FTCA for administrative remedy first.
His problem is:
1. The government isn't acting on him directly, so he has no standing. He could try suing his employer if they force the issue (but he'd lose that) or the employer could sue the contracting agency.
2. The event he's wrapped up about hasn't happened yet. There is currently no such mandate as it has been blocked by the courts. And if you follow HIS strawman, he wants to sue AFTER the courts declare it unconstitutional. Well, that would mean that the event never happened. You can't sue over "might have" just "actual" damages.
3. Any suits filed now would be suspended or dismissed if there is the same case already pending. The federal courts hate duplication of effort.