There are several main factors as to how long Civ IV games last.
1) Game speed setting
The difference between quick (everything is 33% faster build/research) and marathon (everything is 300% longer to build / research) also manifests as an actual increase in total number of turns. (I think "normal" defaults to 500 turns?)
2) Game Map Size
Map size can change the length of victories immensely. Small maps favor conquest or domination victories. Larger maps (with the chance for more science producing cities) lend themselves to (marginally) faster space and Cultural victories.
3) Victory Settings
As mentioned by Marco elsewhere in this topic, there are several different victory conditions. Some of these (Conquest / Religious / Time) can be reached markedly sooner than others (Cultural, Space Race, Diplomatic)
4) How you play the game
Do you automate everything and press "enter" to skip turn after turn? Are you a conquest-minded gamer who needs to micromanage large armies as you expand your empire? There's no easy way to determine how much time a single turn will take.
The bottom line is that a standard Civ IV game will happen over hours, if not days. (Though you may quit at any time, receiving a "Time" "Victory")
Edit: Added "Game Map" as suggested.
Unfortunately, I do not know exactly how much you must culturally dominate a city for it to switch. That number may well depends on your difficulty level.
However, I think I can help answer your implied question: This city is more culturally mine. Why hasn't it switched to my empire?
First be sure city switching due to culture is turned on.
I assume it is, but this is an option you can set before the game, so its worth mentioning.
The threshold might not be 51%
This is what you supposed, and I agree this might be true. Until someone else chimes in with a number (or several for difficulty) lets assume something else is wrong...
Culture switches are not immediate.
The city will usually have at least one cultural revolt first, i.e. both you and the current owner of the city will see an event like "A [Spanish] revolt is taking place in [City]." This is a great sign for you, and means that you passed the threshold; the city would switch to your empire fairly soon if it had no troops in it. This has the unfortunate consequence that...
The current owner can put troops in the city to slow you down
The longer the city is in revolt in the name of your culture, the more likely it is to switch. So naturally, if the revolt is suppressed by, say, more troops, its less likely to be successful. I think with enough troops the revolts can be prevented altogether. I'm not sure of this, or whether a city can switch without revolting first.
Troops are the only way I can remember the current owner being able to fight you, but there may be others options I can't remember, like particular government choices or wonders?
Consider destroying the troops with espionage
If the troops can slow down the process, then get rid of them. You sound like you want to take the city without war, so spies are the option you have left. But, if you don't care whether you go to war or not...
Consider taking the city with military
If I recall correctly, a city that prefers you culturally will be "liberated" rather than "conquered". Unfortunately, this has the same problem you asked about taking the city cultural: how much is enough? I'm not sure, but I believe if they have revolted for you then its enough.
Liberating a city is effectively "this city is supposed to be mine, the people welcome me", I am certain its what happens when you retake a city you founded but lost, and has great effects:
- The city does not lose a population due to being "conquered".
- The people do not revolt, but begin working immediately.
- Buildings are more likely to survive the transition. (Perhaps all of them do?)
- (Does not apply to your situation but...) The cultural border the city had when you last controlled it are restored.
As a final note, if you could liberate a city, and a third civilization takes the city, they immediately have the option of "returning" it to you.
I feel like I've missed something, I will add it later if I think of it.
Best Answer
Yes, increasing the global culture rating will increase the culture produced by each city, which will increase the rate at which your borders grow. With the right buildings, the global culture rating also adds happy people to your cities, which helps them to grow larger.
If you hover over a square, you can see the current balance of culture in that square just above the unit picture. Whoever has the highest total is the current owner of the tile. If your culture that reaches that square is higher than the culture of the other civ, your percentage will grow over time.
In your own cities you can also see a percentage of culture. If you captured the city from someone else, it will start out heavily waited towards them. The higher the percentage is for opposing countries, the more likely a city is to defect. You can't see a percentage of culture in cities you don't own, but it works similarly. If you have significantly higher culture output near an opponent's city than they do, you can eventually get that city to defect to you without warfare.
You need to be careful of capturing a city that is well surrounded by an opponents cultural borders, as it will likely defect back quite quickly. In this situation it is often better to just raze the city instead. You can decrease the likelihood of defection by stationing additional units in the city, but this often isn't enough, and if the city defects you lose the units too. Generally, the best strategy is to capture cities on the outskirts of their territory first, where they will have less culture output. Or be prepared to capture all their large cities in rapid succession. Also think carefully before sparing a civilization, as destorying the civ completely insures that your cities can't defect back to them.