Midnight Clock's first triggered ability triggers at the beginning of each upkeep, not just your upkeep. In a Two-Headed Giant game, an ability that triggers at the beginning of each upkeep rather than each player's upkeep triggers only once during each team's upkeep.

You can activate Midnight Clock's mana ability to pay the cost of its second ability.

Midnight Clock's last ability triggers after one or more counters are put onto it if it had fewer than twelve counters on it before those counters were put on it and it has twelve or more counters on it after.

If Midnight Clock leaves the battlefield while its last ability is on the stack, it won't be exiled.