Early rush or not, the easiest way is still to lib MT and cuir rush everyone left alive. You can do this by expanding peacefully to 7-8 cities and bulb your way to lib, or you can take out one or 2 neighbours with for example HA, then bulb your way to lib. Sure you could also take out your entire continent with HAs on monarch, but that is not exactly easy if you haven't won a military victory before.
The strategy when aiming for cuir rush is to focus on the top part of the tech tree. The wonders you want are usually Great Library and Parthenon, possibly Mausolleum of Mausollos and Taj Mahal. And of course the Oracle if you can get it, which you should be able to on monarch. (On monarch I'd probably aim to oracle Civil Service, but you can go for something cheaper if CS feels hard to get). Then you aim to produce Great Scientists, the first one builds an academy in your best science city (mostly capital if running Bureaucracy) the next bulb Philo, Paper and Education (edu requires two Great Scientists). The biggest techs you need to selftech are Nationalism and Gunpowder. You can either bulb or selftech liberalism once you have Nationalism and Music, then take MT as your free trech.
The lower part of the tech tree you can usually trade for and mostly don't need to research stuff like Machinery and Guilds yourself. If you don't do an early rush you can also usually trade for HBR later when it's time to prepare for cuir wars.
Once you get close to MT, start building some HA or Elephants that you can upgrade to cuirs as soon as you have MT. When you reach MT you can shut down research completely and just focus on pumping out as many cuirs as possible as fast as possible, as long as you get there early enough to finish the map with cuirs. If it's a continents map and you need astro, then you have to continue researching towards that and possibly even get rifling if it takes you long to get to the other continent.
As Gumbolt suggested, start a game of your own, or participate in a forum game (there's a new NC starting today, right?) and people can help you along the way.