So, what's the question exactly?

Is this a plumbing question? or "how ugly is it to physically move this thing" question?

The good news is that you probably have a basement or at least a crawl space on the first floor so you can do the plumbing easily enough. The old plumbing can be capped off.

The larger question in my mind is moving it.

Be sure to measure doorways, etc. through the entire path. This thing is probably 400 pounds of cast iron and if it gets away from you will do a great deal of damage to anyone / anything as it's going down the stairs.

Reinforce the floor on the downstairs bathroom where the tub will go. Double up on a couple of floor joists under it or add some basic pier supports if you are over a crawl space. With water in it the thing will weigh about 900 pounds and I see plenty of older bathrooms where the weight of the partition wall and the tub cause the floor to sag.

Something to think about is taking a sledge hammer to the one upstairs and taking it out in pieces and then buying a new one for the bath downstairs. This avoids the danger of a tub getting away from you AND the cost of refinishing an old tub. You can get a new one for $1000-1200. Messing up your back / smashing your fingers or worse would cost A LOT more.

It sucks. We have done it.

You need a few strong men and a reinforced floor. Cast iron is heavy. Plumbing wlll depend on the state of your home-- has your plumbing been updated? Will the new tub sit with the same orientation to the floor joists so that plumbing can be re-run with ease? Where is your stack and how far away are putting this tub from it?

Stairs will really be the determinant. Many old homes have staircases that are steeper and narrower than modern homes. Measure first and measure well to see if you can get this tub down any stairs without taking out railings or partial walls. That's probably my best advice.