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.

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.

The last ability of Skyline Despot checks to see if you're the monarch as your upkeep begins. If you're not, the ability won't trigger at all. You won't be able to do anything that would make you the monarch during your upkeep in time to have that ability trigger. The ability will also check to see if you're the monarch as it tries to resolve. If you're not the monarch at that time, the ability will have no effect.

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.”

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.