The free month may have to be a code e-mailed to you, so it may take time, too. But I'm fairly certain that things like Cloud Save issues are probably things they're not liable for in the terms of service, so a lawsuit would probably result in nothing. They're just doing the free month to avoid the negative publicity of an issue for a single day that the internet would raise a digital riot over.
You'd be figuratively destroyed in a courtroom by their lawyers once they bring up the terms of the Xbox Live service, and likely no jury or judge would side with you over one day worth of not playing games. Plus, the legal fees you'd incur would cost you much more than just buying a new Live subscription card.
And I don't think you'd have a case against Major Nelson, as he's just a spokesman for the company, not a direct contact who personally promised you a free month.
I'm not a lawyer. I'm just stating what I expect.