This isn't a spiral staircase, but rather a helical staircase. The difference is the lack of the centre pole (As you noticed. It's otherwise known as a newel) and that it has handrails on both sides, whereas a spiral staircase only has a handrail on the outside.
I was able to find this example of what appears to be a similar design, though it looks wider than your's.
A block and tackle (pulley and rope).
A simple fixed pully provides no mechanical advantage. A moving pulley does. Each time you add another moving pulley, you reduce the effort needed to move something. In general every time you double the length of the rope, you halve the effort (perceived weight).
![pulley system](https://i.stack.imgur.com/VMxMy.png)
In the illustration, the pulleys are both doubles, something like these.
One pulley is attached to the stair structure, the other to an eye screwed into a stud facing the stair, or to a stud above and forward on the wall alongside the stair.
The rope is attached to an eye above the wall pulley then threaded to the opposing pulley, back to the wall pulley and so forth.
A block at the foot of the structure prevents it from sliding.
This type of setup can reduce the load to about 1/4 of the actual weight (maybe 100 lbs.) More pulleys and rope can reduce it further, but you get more friction and more chance of tangled lines.
This is definitely a multi-person project. You need help in pulling the rope and in guiding the structure, and maybe someone else to place a supporting brace under the unit when it is in place.
Alternative
To avoid any tie in to the walls, you can get a hydraulic jack very cheaply (maybe $20) that can lift several tons but only about 6 to 12 inches at a time.
![hydraulic jack](https://i.stack.imgur.com/Yyro0.jpg)
You could do this, chocking the unit at each lift, then raising the base of the jack on a wooden block and doing it again.
This requires careful blocking and patience. It also gives you less ability to shift laterally, but it could work.
Best Answer
The number one reason for creaking floors, staircases and furniture is they are assembled using regular nails. The problem with regular nails is the following.
When you put load onto any board in your assembled staircase the board bends slightly and this causes it to slightly pull the nails at the places where the board is connected to other parts. After this repeats for thousand times the nails remain slightly extended out of the part and so the board is no longer tightly attached.
Now every time you put load the board bends and the walls of the nail holes rub against the nails and this causes the creaking sound. When you remove the load the board restores its original form and the same rub-and-creak process repeats in reverse.
So the bottom line is you have to reassemble the staircase using some fasteners that withstand extreme alternating load. I had great success using wood screws for furniture. One answer to this question mentions that wood screws are brittle and so maybe less reliable and so you could use some advanced nails. Anyway you have to use fasteners with extreme resistance to pulling out.
You likely don't need to dissassemble the staircase completely. In most cases adding new fasteners or replacing old ones one-by-one will be sufficient.