The cost can vary hugely depending on location and who does the work, but you can expect it to be on the high side since it involves some non-traditional techniques and a lot of hand stitching, two things that a lot of upholsterers don't do much.
The material from the outside of the chair will help as a pattern but your upholsterer will have to wing it for the inside. It's hard to get 54" wide fabric draped exactly right so that it covers the inside arms all the way to the front corners. I have done three of these to date and I think every time--despite careful positioning and anchoring with T-pins into the foam---I ended up with less on one side than the other and had to really sweat it to get it to cover the same way as the other side.
The other major problem with these is that the frame is styrofoam, which melts on contact with conventional spray adhesive. I ended up covering the entire inside back with strips of muslin fabric glued to the styrofoam with a particular brand of white glue that was made to adhere to styrofoam (there's a website called Glue This to That, or something like that, and even then the only place that had that white glue was Ace Hardware). Then I used spray adhesive on the muslin-glue layer, which worked ok. This was only necessary on concave curved areas to keep the upholstery fabric from tenting across when pulled tight. Convex curves were fine as is.
The upholstery fabric still has to be glued down to the inside back too---not on convex curves, though.
Then the entire perimeter has to be sewn by hand to get it to fit smoothly. There are thin plywood tacking embedded in the foam in places but they are not continuous. The original covers are sewn by machine but I was told that heat was a factor---either heated before putting them on so that they were stretchier, or heated after they were on to heat-shrink them on. I don't know--I don't like vinyl anyway and I do like to do hand stitching so it worked out fine for me.
All of that is more than a lot of conventional upholsterers want to deal with. You need to find someone who likes a challenge or who at least has done some curvy things where fabric is glued down. I don't really keep track of upholsterers around the country so I can't help with referrals.
PS, i'm not a guy but don't worry, you're not the first to make mistake and won't be the last, I'm sure.
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