The Millenium Calendar's first ability triggers during the untap step. However, since no player gets priority during the untap step, the ability waits to be put on the stack until the upkeep starts. At that time, any "beginning of upkeep" triggers will also trigger. Those abilities and The Millenium Calendar's ability are put onto the stack at that time.

If The Millenium Calendar's last ability is countered or otherwise removed from the stack, it will just trigger again immediately if The Millenium Calendar still has 1,000 or more time counters on it.