Note that you need to send your naturalization certificate to apply for passport and you need to original to update your SSN. So you cannot change it till you get the passport/NC back.
Also SSA is a real PITA - takes around an hour in wait time for a 5 minute change. they don't have a separate window for people wanting to just change their status (takes 5 min or less on avg) and people seeking SS benefits (avg of 30 min, one guy hogged a window for entire 70 min that i was there). so depending on the rush, you are in for the most boring time of your life. My advice for SSA is to avoid lunch times and possibly do it first thing in the morning.