When Chronobot enters the battlefield, switch upkeep steps with target opponent for as long as it remains on the battlefield. (You take your upkeep on their turn, and they take their upkeep on your 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.
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.
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.