Sorry, it took me a bit to finish reading Ultimate Campaign. All right, a comparison of the kingdom building rules in "Of Cities and Kings" from Rivers Run Red, the second issue of the Kingmaker Pathfinder Adventure Path, and the kingdom building rules from the Pathfinder RPG hardback Ultimate Campaign.
They are basically the same rules, slightly expanded. What is 10 pages in Rivers Run Read is 30 pages in Ultimate Campaign, plus some pages of optional rules.
Some of that is more options - like in terms of Leadership Roles, Rivers Run Red has 11 and Ultimate Campaign has 14 - and some of it is padding; for example the Councilor entry has the same mechanics but 6 lines of fluff verbiage rather than 1. So that additional page count is part more stuffs and part blabbering (though if you were confused by the sparsity of explanation in the Kingmaker version, that's not necessarily bad).
The "Improvement" phase gets renamed the "Edict" phase and other such, but in general you're getting the same system with a bit more of each of the lists-of-bits. There are some more buildings; you can do terrain improvements (waterways, bridges)... What it doesn't appear to have a bit more of is playtesting. Same system, most of the same holes. They did do bolt-on fixes to plug some of the most well known holes, like in the Income phase you can no longer sell magic items created by your city (no real in-game reason given, just to stop the "infinite monies exploit" popular among Kingmakers). They also took the Economy +1 and left only the Loyalty +1 benefit on graveyards, to avoid the Necropolis Gambit. So the various costs and DCs are left 90% the same, with some tweaks.
In general you are getting the exact same system in Ultimate Campaign (same grid layout page, same mechanics) with some additions and some cursory changes to prevent the most well known of the exploits in the previous system.
Some quick searches show me that the internet is full of historical accounts, movies, TV-shows, and stories about kings, rulers, and other such nobility and what they do during their day. I am sure you are aware of this as well, so I understand that you want an answer in the context of a role playing game.
The King can do whatever he wants
Regardless of what is expected of him the character can always decide he just wants to go riding through the woods, or go down to the tavern and get drunk, or take up pottery lessons. But that is not something new in role playing games, it's just important to make a note of it. Moving onto...
Everyone wants to meet the King
As a ruler, everyone will want to meet with you, talk to you, and get your feedback, authorization, or approval. From commoners seeking an audience, merchants seeking trade, foreign dignitaries, and your own councilmen. This takes up time, and your day is probably full of scheduled meetings with important people, while those below you take care of the less important people. Which leads to...
You need to lead
Being a leader means you actually need to do some leading. This can be as simple as just writing decrees from your bed, or going to council meetings and discussing with your council on the affairs of the kingdom. It's all fine that you have allocated and delegated most of your power down to various minions - but you need to keep guiding them. They will have daily, weekly, monthly, and yearly reports for you. You can further direct your kingdom by getting updates on the various aspects within your kingdom - and these meetings with your minions are how you get that knowledge. You might want to change policies, or even change the minion taking care of the policy. But that is also part of...
Secrets!
What good is a king without some secrets? whether its your spies, your mistress, or even your spies mistress - there are some things that a King has to deal with that shouldn't be made public knowledge, hiding these things requires finesse! You need to be able to slip away from people, arrange private meetings, send secret letters, and all this with your schedule full, and people everywhere looking for you. So naturally...
You need to have fun!
What good is being a king when you can't hold a royal ball? Or a feast? Or a tournament? Yes most of the work is done by someone else - but you are the vision behind it. You decide what and where and who, and someone else will build it and manage it. But you need to make sure that you have enough money, which leads to...
You're on a budget!
You are the King yes.. and you are the one in charge.. but only as long as everyone is in line... which usually requires money. Money that you collect through taxes, trade, tithes, and adventures. But you also spend this money - and it is your job to either balance this spending or have someone else do it. And if that doesn't work out for you, you might need to ask for help...
Dealing with other Kings
You will need to send emissaries, or even travel yourself. You will be trading, warring, politicking, marrying children to each other and getting into lots of untold naughtiness. That is a fact.
So if you want to know what a King does with his time? I answer you - What time??
Best Answer
There are two ways to approach a "kingdom building" campaign. One is from the political side, and the other is from the build-and-supply side. The former focuses on social maneuvering and only touches lightly on logistics. The latter focuses on logistics and only touches lightly on social maneuvering.
If you're looking to run a dramatic political campaign, my favorite reference is Dynasties & Demagogues (for 3.5E) by Chris Aylott, published by Atlas Games. It gives solid advice for running a political campaign without it getting bogged down in boring stuff. It adds politically-oriented spells, personality feats and other feats, leader-based prestige classes, and a political maneuvering system that is a lot like social combat.
If you're looking to run a campaign where characters focus on building strongholds and dealing with logistics, my favorite reference is Stronghold Builder's Guidebook (for 3.5E) by Matt Forbeck and David Noonan, published by Wizards of the Coast. It focuses on the stronghold itself: building it, defending it, mapping it, and so on. That may be more limited than you want for a "kingdom building" game.
More like Birthright is Strongholds & Dynasties by Paul Tucker, published by Mongoose Publishing. It provides a system for governance, economy tables, rules for building strongholds, a mass combat system, and so on. It's pretty complete and solves many of the same problems as Birthright, but in a different way.