If Chronobot enters the battlefield during an upkeep step, its effect doesn’t affect that upkeep step. Similarly, if Chronobot leaves the battlefield during an upkeep step, its effect doesn’t expire until the upkeep step is over.

While it’s your upkeep during your opponent’s turn, you’re the active player. However, any rules or effects that check whose turn it is will still know that it’s your opponent’s turn.

Untapping your permanents occurs in the untap step, and drawing a card occurs in the draw step. Neither of these are affected by Chronobot’s theft of time.