If the triggered ability that causes the monarch to draw a card goes on the stack, and a different player becomes the monarch before that ability resolves, the first player will still draw the card.

The game starts with no monarch. Once an effect makes one player the monarch, the game will have exactly one monarch from that point forward.

Being the monarch carries two inherent triggered abilities. “At the beginning of the monarch's end step, that player draws a card” and “Whenever a creature deals combat damage to the monarch, its controller becomes the monarch.”

Abilities that trigger whenever you “become the monarch” trigger only if you aren't already the monarch. For example, if you are already the monarch as Custodi Lich enters the battlefield, its last ability won't trigger.