No, the multiplayer online component doesn't work that way, so there is no peer-to-peer games as you describe.
The online component is in two parts. Caesar's Challenge is a series of maps played over a two week period. They all have singular "win criteria", ie. population, denarii after x months, high score, etc. Each "season" lasts two weeks at which time the same maps are run again but with different criteria. Surely, at some point, the current maps will be swapped with new maps and then we'll play all them by the different criteria.
The other component of Caesar 4 online play is the Empire. Again, this is a set of 10 maps (more released over time?) which are played with primarily population in mind, but it is the player's empire and if they want to make a prosperous empire, or a minimalist empire, or a very appealing empire, whatever, it's up to the player.
To play either, you have to remain connected to the internet while you're playing the mission. Once you've completed the requirements, you'll be given an option to upload your score, which will add it to your own personal profile page online and enter you in the appropriate contest.
Good luck.
[This message has been edited by homegrown (edited 11-04-2006 @ 11:46 AM).]