I'm reasonably certain that the USPO won't let you live in a post office box, and the NY DMV knows what your friend's real address is. You can use one as your legal mailing address, but it's definitely not the address the lottery will use in releasing information about what town you live in. If you live in West Podunk and don't want the lottery to tell people that's where you live you'll have to actually move somewhere else. So far I haven't needed to find out, but I'm guessing that if you live in West Podunk on March 31st , move to East Podunk on April 1st, and then claim a lotttery prize on April 2nd the lottery will release your address as East Podunk, without caring (or investigating) when you moved there or where you moved from. I'll note that putting an incorrect address on the claim form may constitute fraud, forgery, and/or perjury.
I'll also note that your name and where you bought the ticket may be more important to any strangers who try to find you. If your reported address is far from where the ticket was sold I'd look in both places if I wanted to find you. If your name is Bob Smith strangers will have a hard time figuring out which Bob Smith if you live in NYC, but if the ticket was sold in Upper West Podunk and there's a Bob Smith in the phone book for Podunk Center I'd put him on my list of suspects.
The only problem with not being anonymous from the start is having your picture published (ever see Joaquin Phoenix on Letterman?) and having people who already know you know you won. Assuming you plan on enjoying the money you won, it's going to be hard to keep it a secret, anyway.
The good news is that no matter what your name is, it's fairly easy to disappear after the press conference. Even if your name is Ebenezer Wilfred Scrooge the 4th, there's no reason it has to be connected to addresses on anything other than your DMV record, your tax returns and your social security info. You'll need a legal address for those, but there's nothing to say you have to actually spend much time there. Just about every thing else can use an address for a lawyer or other representative. Most of the other things won't need your real name, either.