It was long touted that, since there was no year 0 in the AD system of Dionysius Exiguus, that the new millennium would start on 1 January 2001.
However, the calendar turning over to 2000 became the popular celebration of the millennium, so the popular New Year's Eve celebration was 31 December 1999. So Newman's party would be technically right on time, at least according to the previous conventional wisdom, if the hotel made his "Millennium New Year" reservation on 31 December 2000. Thus Newman's reservation does not conflict with the party Kramer is planning for 31 Decemeber 1999, but will be perceived as one year late and "quite lame" according to Jerry, as the majority of people will be celebrating the new millennium on 31 December 1999. Jerry's "one year late" remark was simply to annoy Newman by showing him that his reservation doesn't match his expectations.
Newman believed the millennium to commence 1/1/2000, as he told Kramer: "...he's out of my life, starting in the year 2000. For me, the next millennium must be Jerry-free!" Furthermore, it was established that Kramer's Millennium party was scheduled for 12/31/1999, and Kramer becomes upset upon learning that Newman's party is on the same date. Thus, the invitations for Newman's party indicated 12/31/1999, but his hotel reservation is for 12/31/2000, one year later.
1. If Newman's party is on the ACTUAL millennium (1/1/2001), that's a year late because most of the world ended up celebrating 1/1/2000.
2. Newman's party was scheduled for 1/1/2000 (see the part about the invitations, above) but his hotel reservation is for 1/1/2001.