This quest is slightly more fair than most, pitting against only a single enemy, the formidable King Richard the Lionheart. However, you start this quest with only 2000gp versus his 4000. This is a very asymmetric map, with Richard holding the high ground in the northwest while you are on low ground in the southeast. The most peculiar thing about this map is that there is absolutely no iron whatsoever so any swordsmen, knights, macemen or pikemen will require you to buy in iron or weapons. Lastly, you start the game surrounded by several lions that will make farming and woodcutting impossible without clearing them. All this seems rather daunting, but if you act fast there is a cheap "unfair" way to make this map easier.
First things first, you want to go into your options and slow game speed to 20, as the first few moments can be critical.
Begin with building your Trading Post, Stockpile, and Granary (set to 1/2 rations for the moment), before building a square tower as far away from you and as close to Richard as possible (it should be on/near the edge of the grass). Place some stairs, braziers, and a mangonel before building an Engineer's Guild. Conscript a couple engineers and race them and your starting archers up into the tower. Your archers will now have the range and safety to be able to pick off most if not all of the lions given a bit of time. With your mangonel manned, look towards Richard. Unfortunately, you are too far away to target anything juicy, but he should have built a gatehouse that is JUST within reach (if it is not yet built, move on to other things for 15-30 seconds and check again).
With your mangonel shelling his gatehouse constantly (and with scatter damage hitting his barracks, granary, and other goodies), Richard will go nuts and be completely unable to defend himself or set up. It is imperative you hit the gatehouse as soon as it spawns, as every second counts. He will attempt to send swordmen or catapults to retaliate, but your archers should keep you safe from harm. Go ahead and set game speed now to whatever you prefer.
On to domestic affairs. Unfortunately there are an assload of trees and little space for farming. Place an apple farm or two where you can, and spam woodcutters around the oasis. You'll need those trees cleared so you can get some proper farms going mid-game. Meanwhile, build as many quarries and ox tethers as you can support, probably with some woodcutters to chop away at the foresty nuisance surrounding it. Use excess coin to buy in a bit of meat and cheese to cancel out your half rations. Make sure your popularity doesn't fall below 50; raise rations or give bribes for a while if it does.
Now you have some time to breathe and plan out your path to victory. Richard can't do much to you except rebuild his buildings, or send a couple catapults or swordsmen your way. If you feel insecure, buy in some crossbows and leather when you can afford it and get some crossbowmen into your tower to take on any swordsmen he might send your way. I also suggest destroying the bottom stair of your tower so it can't be entered by enemy troops.
Since all you really have on this map is stone, wood, and (later; much later depending on how many woodcutters you have) some farmland, you'll have to get creative to make money. Stone you can sell for some decent money, but you're not getting a whole lot of it at a time. What you are swimming in is wood, however. The unfortunate thing is that wood sells for almost nothing, so we'll have to upgrade it into something better. My approach was to sell my stone and use that money to produce a horde of fletchers. Selling bows is worth far more than untouched wood, and crossbows sell for even more than bows (I suspect crossbows take more wood/time to make, but this may be incorrect).
After the woodcutters clear up enough farmland, destroy the huts and build some more woodcutters near other trees and fill up the farmland with farms of your choice. Once you have enough money to start making plays, how you defeat Richard is up to you.
Since most siege weapons require constant babysitting and stone while the mangonels don't, I replicated my tower of pain until I had three towers bombarding him. They stopped firing once I took out the gatehouse, but I would target the nearby walls until he rebuilt, to ensure he was constantly being trolled. Then I placed some dairy farms and tanners, bought in a wad of iron and build blacksmiths to produce maces. I build a mercenary post and alternated between recruiting macemen when I had the weapons/armor, and arabian swordsmen otherwise. The macemen can punch a hole in the wall if Richard has repaired recent mangonel damage, and alongside the swordsmen it should be an easy win.