The Lost City quest is one of the easier boss fights, but with a catch. You cannot bring any gear. This means weapons, armor, magic robes and ammunition. However, you may bring runes, potions, food, fletching gear (unfinished arrows and unstrung bows) or just use good old fashioned kicking-to-death.
For the iron man and hardcore game modes the Lost City quest shop is the only shop to get dragon daggers, maces and longswords.
How to start:
Start off by going south of the bank at home, to the wall of quest NPCs.
Go to the Monk of Entrana on the far right side.
In his dialogue options, select "miniquest"
You will find yourself in a cavern underground. You may now fletch your weapons, if you chose that route.
Further down, you will find several level-24 Zombies with 30 hitpoints each.
You will need to kill the zombies until you get a bronze or iron axe drop.
After getting your axe, continue further into the cavern. You will encounter some greater demons, but head immediately south of them and ignore them.
You'll arrive at a shimmering Dramen tree.
When you attempt to chop down the tree, you'll be attacked by a level-101 Tree spirit. It's very easy to defeat with protect from melee, or just a few pieces of food, but be warned it can hit up to 11.
Congratulations! You've completed the Lost City quest! You now have access to the Lost City Shop!
The shop:
Dragon longsword: 60,000gp
Dragon dagger(p++): 22,400gp
Dragon battleaxe: 121,500gp
Dragon mace: 30,100gp
Hunters' crossbow: 1,690gp
Long kebbit bolts: 570gp