Just moved from Noble -> Prince myself.
Pay attention to the things you didn't before. Almost everygame I play now, I take FOREVER to make my initial 50-100 turns. Plan ahead and stick with the plan. Set mini goals for yourself to reach and reach them. A huge help that allowed me to catapult from noble to prince was better management of building queue and more focused city specialization.
Trade with AIs alot. Not just to trade, see how many cities they have, if they got 1 more than you, get another one. Check the power chart every 10-20 turns. If your not at the top 20%, get back there. Don't shun the idea of setting EVERY city to make military units at some time. WHen your low in power, whats perferred? You have to wait 20-40 turns to get back to the builder mode, or Monty takes 2 cities.
Plan wars around key techs and units. If I see I'm going to get artillery, or cannons, or what ever kick ass unit before someone else, when the tech is a few turns away, science slider goes low as hell for upgrade gold, get the tech, now 20 cats turn into 20 cannons and the war is on.
In Noble and under, bronze working & chopping if you have forest around you can give you one hell of a leg up at the start and give you 2 cities very quickly. With enough forests, you can have 3-4 cities when the AI has just their 2nd. Pre-chop forest till they are almost gone. Do a few like this when the worker has nother else better to do more important. If you don't need that cottage in the next 20 turns, pre chop. Now when you want your worker/settler, pick it, chop the pre-chops and now you have a worker/settler in about 10-20% of typical build time and your city was stagnate all that time.
LIttle things this like this make a HUGE difference. 1 extra commerce coin in a worked tile means your science budget could be almost double at the start. Ever wonder why some games, Bronze Working is like a 30 turn research and some others, its maybe 19. This is why, got some extra commerce your working early on. if you intend to chop a worker/settler, DON'T let the computer manage the tiles as it will focus on food & hammers. if ur chopping, focus on commerce during that time.
All little things like this add up. Mind you not an expert, no where near it, but I feel ya man. I had to work my way up through the difficulties as well.