Clearly your experience is different from mine -- Sprint and Samsung were absolutely horrible about getting Android OS updates out to my old original Galaxy S when I had it. I had to wait ~9-10 months for an update and it was not unusual for my phone to be two versions of Android behind the current one. Based on that, I'd say not to expect it before, oh, about August 2014. But elsewhere around here people have said that Samsung has gotten better, which I hope is true, but given my past experience, I'll believe it when I see it.
However, I've read a few things regarding KitKat for the Note 3 -- from it could be out as early as this year to it may not come until after Samsung releases it pre-installed on the Galaxy S5 sometime around spring 2014. I'm not sure there's much good information out there about it, since neither Samsung nor Sprint seems to be saying anything official. FWIW though, I've read elsewhere that KitKat for the Note 3 is supposed to be in testing already; I think those posts all quoted SamMobile as the source (one of their tweets, I think). If that is true though, it suggests that we could get an update sooner rather than later.