First get to a good vantage point on the far side of the upper level. Once there, you'll need to do a similar thing as to how you broke the large door down in the mine. Make your way over there, using your rope arrows as needed. You'll have to use the trebuchet on the other side for your second shot. Then, turn it to the gate and the debris will become dislodged. Then, go to that platform and use your rope arrows to connect the pole to the debris jammed in the trebuchet. Once there, aim it first at a nearby ledge with a roped pole. You'll need to swing and climb the structures to get to it. Once cleared, make your way through the door on the level you fought on and go up to the trebuchet. Otherwise, use your shotguns for the heavily armoured ones or your assault rifle to break their armour and your fire arrows to set them alight. Use poison arrows and Greek Fire when enemies are in groups. You'll have to fight a few waves of enemies here, so try not wasting ammunition. Head left to the next gate where you'll get armour-piercing arrows. Make your way back and go through the gate. Get up to the trebuchet, kill the guards, and use it to break down the gate. Once in the far right corner, use your broadhead arrows to create a climbable path up the walls. There will be lots of heavily armoured soldiers, but most will only use melee attacks. Once there, though, you'll need to head back and up to the right where the trebuchet is. Use the zip-lines to get all the way across the old city to the gate, making your own if needed.

The lost city tomb raider map