Regular elevators generally operate by use of a very long cable and counter-weights.
Obviously a kilometer high cliff is far too high for any conventional rope, and steel cabling is perhaps outside the scope of your civilisation's technology.
However, a very clever artisan came up with a system of toothed tracks and gears which allow an elevator to climb any height as long as it has power.
A pair of oxen in a large hamster-wheel can provide that necessary power, which is translated through gears to the track and hauls the entire carriage up the cliff at a steady pace.
It may take a while, Oxen are not inclined to go quickly, and what with gearing the oxen are going to be travelling a lot further than the kilometre of the climb itself.
But you'll get there in a few hours, which is better than sailing downstream and walking a five or ten kilometres uphill.
"But wait! What's stopping the hamster wheel from spinning like a blender and killing the oxen?"
I hear you ask?
Worm Gears my friend.
A Worm-Gear is essentially a rod with a helix-shaped gear running down its length. Imagine a screw-thread without the pointy bit or the bit for the screwdriver on the other end.
With this, the motion is one-way. A cog cannot turn a worm-gear, but the worm-gear can turn the cog.
Consequently, with enough torque from the hamster-wheel and its oxen (and probably some clever gearing) You can travel up and down in complete safety.
The oxen can even get out of the hamster-wheel and be changed for fresh oxen at the end of each journey without the need for brakes!
The other concern of course is wear on the tracks themselves.
Most likely you'd want to make them in sections as pieces of bronze or steel, but really if you make them chunky enough, wood will do fine.
You would want to have the mechanisms redundant and applying their forces across all four corners of your elevator too.
Ultimately, it'll have to be quite a large elevator. Think in terms of a structure comparable to a small building, with a hamster-wheel of oxen and gears in the lower half, and the cargo/passenger area on the upper level.
Of course, If it takes hours to get up and down you'll definitely want to have multiple elevators operating in parallel.
Most likely they run on a rigid schedule, perhaps with three or four elevators running, one might begin its journey each hour, every hour, and so workers in the port may live in the city above and commute each morning and evening.