Random theoretical question...

I guess it depends a bit on the situation the city state is in.

If it were an island city state, I guess some form of democracy. Maybe every five years eligible people vote for a Governor that runs things and can make and change laws (except the one about a five year term).

Harder if the city state is part of a larger nation or body of land. Since this is ‘travel back in time’ and not ‘made up world’, wars have been started over differing types of government. Maybe some kind of oligarchy. A council of city elders or experts (a general and a farmer, for instance) each representing a section of the city and they lead or rule by consensus or majority vote between themselves.

I would adopt the words of a very wise, yet fictional, woman as my guide.

"All shall love me and despair."

I would not plan for my succession as that would be admitting my mortality. I would have anyone who broached the subject thrown into the bottomless Pit of Immortality so they would have ample time to contemplate their poor life choices. There would also be free pie every Sunday.

Whatever is in season. Funds are not unlimited and Momma needs new frocks.

This thread weirds me out a little, because I have pondered this exact scenario for as long as I can remember.

My most relevant thought after many years of pondering it is that, regardless of what decisions I or anyone might make, adjusting to life without plumbing and electricity would suck giant donkey balls.

Damn aristocrats, using paper from them there learning books to wipe when all we peasants have are leaves!

REVOLUTION!

This thread contains some people who just joke around, and it contains some people who seem to be highly delusional.

Life in medieval times was very, very hard. Basically, everybody worked very hard to just stay alive. And that is the normal, natural human condition. It is also the normal human condition that most people failed at that. Despite that many people, of those reaching sexual maturity, managed to have double digit number of children (but never alive at the same time, of course), populations seesawed up and down, but stayed small.
There is an extreme lack of prosperity, and prosperity is needed to do anything at all, other than just survive.

The economy looks like this: Let's say that those things which have an intrinsic value, have something we will instead call "worthness". Only food, water, shelter, weapons and tools have worthness, nothing else. (I've split weapons into weapons and tools, as that probably fits people's concepts better).
Other things, only have value, if somebody is willing to trade something for it, and that trade chain must start with things that have worthness. So let's rephrase that for clarity. Anything else, has value if, and only if, somebody is willing to trade items, that have worthness, for it. For other activities, there has to exist an excess of worthness.

Yet, precisely because prosperity is scarce, and because prosperity can be gained by taking other people's things, and maybe their time and hard work as well, as slaves, the very first and most important thing that any excess prosperity must be spent on, is military. And this is the raison d'etre for feudalism. Its only original purpose. Giant Ambush Beetle is quite wise in focusing on military. However, there are limits to what can be afforded without destroying the population under taxes.

 - Oh, but I'm sure that always depends entirely on the context. 🧐

Don't worry, I'm not going to try to moderate threads. 
Preamble: Noun. a preliminary or preparatory statement; an introduction:
Example: "what she said was by way of a preamble" ·

It's feudal to continue this conversation, why don't we try having a renaissance of good discussion and try for some enlightenment? This current topic is baroque and there's no fixing it.