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.
With a little trial and error the stain can be matched, color wise. The character of the grain never will, but that minor detail will not stand out. My windows are stained pine with stained poplar trim surrounding the sash. The wood that makes up the sash does not have the size to make it a problem with the other specie of wood that is in the area. Do make a test sample to get the color adjusted to the right color. Use white pine for that too needless to say. I have made samples... got what I though was a good match, and when the finish was applied, it was still off. I have adjusted the color further by adding a little stain to the finish. In this case, I am using oil based finishes. I do not have any experience in doing this with water based finish and stain. I would think the effect is the same though. Do not add no more than 20-25% stain to the finish to tone the color of the finish. I am told it will make the finish too soft.
Best Answer
I have remodeled quite a few Victorians the ones with pine or fur treads were usually stained with a honey or amber colored stain then polyurethane, I used varathane a solvent based poly, I have been told the water based polys are as good but my limited experience with water based was not as good as solvent based.
I use very expensive brushes this is the real trick to a nice finish with poly ($20.00 is normal for a good quality brush) to make the brush last years you will need a container that you can put some solvent in with the brush and seal it, after cleaning most the poly out, put the brush in solvent at least 3/4 of the bristles covered ) Seal the jar next time you want to use a day week or year later take the brush out, shake out the excess solvent seal the jar up and start applying , when you mix your poly stir never shake (unless you have a vacuum top to pull the air bubbles out). When done clean the brush add some fresh solvent to the jar and put your brush in. Do this if even just going for a lunch break once the poly drys the brush is toast.
Poly works best with a medium heavy coat and let dry completely, light sanding between coats for awesome wood grain. First I would stain the entire staircase then 1 coat on the entire staircase. Since you want a carpet runner I would then put down 1-2 more coats on each end an inch or 2 inside where the carpet runner will go. This really brings that old growth wood grain out.