To clarify, this staircase is entirely brick, correct? If so, you absolutely don't want to use concrete. Concrete is harder than the brick and will eventually damage the bricks.
You want to use mortar. If it's an older staircase, it's likely sanded mortar (mortar + sand).
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).
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.
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
OK you have something that probably has a good deal of weight on it with no proper header or support over too far of a range.
In my horrible picture below you need 2-3 vertical 2x4s and bottom plate in blue. Maybe you could space these out with the right size header - like framing an exterior door. But that may carry the vertical load better but it may not satisfy the horizontal load - someone on the stairs. So while you can try putting a header on there, it could still creak...
Also in black you could easily add cross bracing and over time this will keep screw and nails from loosening, making a quieter set of stairs.