This one is a pain in the ass, as you are but a lone figure on top of a lonely hill about to be screwed up the butt by three Sultans. Not only this, but you are trapped within a claustrophobic nightmare that is terrible for a big economy strategist. Oh, and, did I mention, they each have 8000gp while you have 2000? Yeah, this is gonna be rough ride, but with some ingenuity and no small amount of luck this map can be conquered.
This setup is going to require some VERY fast maneuvering to keep you alive so for a while you want to go into Options and set game speed to 20.
The trick here is that since the map is small enough, a strategically placed tower-mounted mangonel can actually shell an opposing lord, as well as a few goodies. Unfortunately, we'll need enough archers to fight off any bowmen that try to kill your mangonels, AND enough crossbowmen to repel arabian swordmen attacks...TIMES THREE. Obviously on 2000gp this is impossible so we're gonna need to make due and do a lot of praying.
Start up the map and you'll hear that your keep is enclosed. Neat! You are protected by two concentric rings of natural rock with two openings apiece, which is already defended by a gatehouse and a (dangerously thin) layer of wall each. Unfortunately there is nothing stopping these guys from sending a few waves of arabian swordsmen to break your puny walls and murder you horribly. That is until you use some cheap tricks.
Build up your Trading Post, Stockpile (don't build it out all the way as this map is VERY claustrophobic and you need every inch!), and Granary set to half rations.
Quickly place an Engineer's Guild and recruit two engineers. If you're fast enough that the game hasn't given you your allotment of stone yet, place an Apple Farm and Dairy Farm on your nearby grassland. Once you have 40 stone build a stone tower WITHIN your innermost circle as far out as you can facing the southwest Sultan. Build the single brick of wall needed to build stairs and race all your archers up there. Build 2-3 braziers and a mangonel and race your engineers up there to man it. Once it's manned, target the southwest lord himself (if you can't pick him out like I couldn't, aim at something juicy like his granary). This constant barrage of destruction will piss that Sultan off, every so often kill some random troops or (occasionally) buildings, while your archers slay any civilians, slingers, or bowmen and provide a (meager) defense against swordsmen waves.
This defense will serve against all but swordsmen waves from the southwestern Sultan, and will also partially defend against the southeastern Sultan (since he will attack you from your southwestern gatehouse). This build will likely not kill many swordsmen, and you have no answer to the northeastern Sultan, but it's better than nothing.
Now we need to get your keep going. Place one Dairy Farm and Apple Farm if you haven't yet, and squeeze in two Wheat Farms as well. Our goal, since we cannot feed our people much, is to feed them a variety to slow popularity loss. Place no more than ten woodcutters around your forested farmland so you can utilize that land as soon as possible. Place three or four hovels within your outer ring, just within protection of your southwest gatehouse (I did this hoping that if swordsmen broke in if they attacked the hovels it would buy me time. Alas they never targeted them, which good news make them safe there). Build a pair of Quarries, one ox tether each, as well as a couple of iron mines within your inner circle.
At this point you'll have a very small amount of gold left, as well as little stone or wood. Don't sell off either unless an emergency appears. If you can, buy 5-10 meat for food variety and once your wheat farms return with wheat build one mill and a pair of bakeries.
Sell iron as soon as you get it; use wood to build hovels, iron mines, and bakeries as needed. Stone you want to hoard for the moment, as you will need it soon.
Now you face a dilemma. If you don't get some crossbowmen facing southwest soon, you will be steamrolled by an inevitable wave of swordsmen. If you don't get another archer/crossbow filled tower facing northeast, the same thing is going to happen WITH IMPUNITY. Your thin walls will make that happen so much easier. Which Sultan makes a first move will help you decide, but either way it won't be pretty.
IN CASE OF EMERGENCY: If a wave of swordsmen come from one end or the other you're going to be in deep trouble. Though the northeast is undefended, both southern lords will attack from the south; pay attention to both. Once you see or are notified to an attack of swordsmen, rush your starting swordsmen and knights to the site to try to stave them off. Quickly build a mercenary post and a hovel if necessary, and poop out as many slingers as you can afford. Short ranged as the bastards may be, they pack a decent punch against armor. Once they breach your outer walls the Sultans tend to send their men at your eastern inner gatehouse, so send your slingers there. If the attack is coming from the northeast I would keep your swordsmen and knights near the eastern gatehouse because without the slingers they'll have no fire support. You can if you're quick enough and possess extra stone try to erect a hasty wall a few layers deep towards whatever direction they're heading as well to give you enough time to pelt them to death.
If you find yourself with excess wood and gold, build as many fletchers as you can afford. Whether you set the to bows or crossbows matters little; you will need many of both, and soon. If you can, building a tanner or two near your dairy farm will save you some coin in the long run recruiting crossbows.
Once you have about 40 stone you have a hard choice; build your northeastern tower or try to fill in your already thin (or breached) gatehouse walls? The choice is yours but I would recommend the second tower as first thing. Build it much as you did your first tower, within your inner circle facing Sultan's keep. Recruit two more engineers, build a mangonel and some braziers, and get the engineers and half your archers up there.
As soon as you can physically afford it after this, you want to build armory and barracks, buy (unless you have already produced) ten crossbows and leathers and recruit all ten crossbows, putting five each at both towers. This will not be a perfect defense but this will give you a LOT of power to cut through swordsmen.
With both towers built you should have room to breath (figuratively, in this cramped hell of a map). Don't forget your affairs; once you have grassland space destroy woodcutters and place more wheat or dairy farms. Build as many iron mines in the outer ring as you think you can defend from attack, and build enough hovels to support however many mills and bakeries you need to sustain yourself. Hopefully your popularity hasn't dipped below 50 yet; if it does, go up to full rations for a while; eventually you will be able to sustain this.
Build fletchers and tanners to cram your towers to the brim with guys. It would also do you well to build blacksmiths and/or armorers for two reasons: if your preferred playstyle relies on macemen, swordsmen or the like you will need to build up your weapons to save money. Alternatively, if you don't use metal gear you can sell the produced armor/weapons for more gold than the worked iron would sell for by itself.
Once you can, you should thicken all your existing walls and build a third tower inside your outer ring facing the southeast Sultan and cram it with a mongonel, archers, and crossbows as before. Now you should be pelting all your enemies at once, with the reach of your archers slaying nearby civilians as well. The Sultans will have spent thousands of gold sending futile waves of slings and bowmen your way to be ripped to shreds, and at least one of them (likely southwest Sultan) will be poor and easy pickings.
How you finish off each Sultan is up to your individual playstyle, but you should target either the southwest or northeast one first as they are easiest to reach and are in more dangerous proximity. I used my existing tanners and blacksmiths to equip a large force of macemen, and while I was building my army I sent half of my existing archers/crossbows away from my tower to stand on the southwest gatehouse, close enough to shoot at Sultan's keep itself, then sent the second half once the first half started shooting again. Once I had a wad of macemen, we rushed forth to slay him. You should probably be able to take the other two Sultans out easily with the same (or slightly replenished) force, but if you prefer to drag things out, now you can spread your legs with some open land. Build whatever you want, just keep in mind to make sure any buildings are protected by covering fire.
The last two should fall relatively easily.