Extractor pins are pressed in pretty tightly. You'll likely end up breaking or severely bending a regular pin punch while trying to break the extractor pin loose.
Armorers are taught to break the extractor pin loose by using a dedicated starter punch of the correct tip dimension (think 1/16" pin punch which has been broken off down to the bottom of the pin shank, leaving the tapered end as the new 'tip', which is how a lot of 'extra' starter punches end up being made
). It's only going to be used to break the pin loose and start it moving.
Resting the slide on a nylon armorer block is handy, as long as the pin is located above a hole in the block, although the basic wooden wedge serves decently enough (ditto making sure there's open space below the pin).
A 4 ounce ball peen hammer is used to start the pin moving. Some of them can be rather difficult to start moving. (We need an 'understatement' smiley.)
As you mentioned, the pins are removed top to bottom and reinstalled in reverse order, bottom to top. Once the pin has been broken loose and is easily moving a regular 1/16" pin punch can be used to finish moving the pin downward out of the slide. Some pressure against the side of the extractor will help prevent the thin pin punch from being bent once the extractor pin clears the top slide hole and the extractor spring is free to try and shove the extractor out of the slide's recess.
In older armorer classes a needle reamer used to be included in the tool kits to clean up any burrs inside the lower & upper pin holes, although the newer production tolerances have made that mostly unnecessary. Care had to be taken not to get too aggressive with the needle reamer or else the holes would be hogged out and the slide would no longer retain the pin tightly enough. S&W doesn't offer (or use) any oversize pins if the slide holes are improperly enlarged.
Reinstallation of the pin can be a bit tricky when it comes to aligning the extractor hole to line up with the lower & upper slide pin holes, and then holding the extractor pressed inward against the extractor spring's tension while starting the pin into the lower hole and then through the extractor hole into the top hole above it. It doesn't take much force if everything is properly aligned. I normally use a plastic mallet to tap the pin back in until it's entered the top slide hole (above the extractor slot) enough to hold it, at which time I switch to a large diameter steel pin punch (which will fit inside the U-shaped channel in the slide which surrounds the pin) and a ball peen hammer.
Care has to be taken to make sure the extractor pin is fully reseated up inside the hole when putting the slide back together. If the pin is not flush with the slide on the bottom it can stick down and drag on the alloy frame. Not a good thing.
An small offset file was also sometimes used to clean up any steel burrs raised during reinstallation of the pin, at the bottom, for much the same reason.
Naturally, even though this isn't a terribly complicated process it can easily result in damage to the slide and/or the pin. The pin is easy to replace but the slide isn't.
Probably why S&W doesn't recommend anyone but a factory trained armorer, a licensed gunsmith or a factory technician perform it.
I'm certainly not recommending anyone not trained and familiar with the procedure try to do it, but I thought an explanation of why was politely in order in response to your question. Disclaimer done.
Armorers are told that it's not recommended to remove an extractor (and pin & spring) except for repair purpose in order to avoid any unnecessary risk of damaging anything. The pistols used for training armorers for extractor repair are commonly called the 'extractor gun' as far as the factory provided instruction guns (and removing rear sights) in class, and the extractor pins become rather loose after repeatedly removals, as you might imagine. The first time an armorer tries to remove an extractor pin in an actual duty weapon it can be a surprising, and frustrating, experience.
Not nearly as easy as it was in the class.
I don't suppose the company doing the refinishing has a gunsmith or similarly trained person available to do this for you, do they ... presuming you're using a company to refinish the gun?
BTW, depending on how old the 5904 is and/or how much usage its received, this might be a good time to check the tension of the extractor spring (as well as make sure the extractor hook is free of chips, etc). Unfortunately this requires a force dial gauge to check the tension, and the standard tension range differs a bit depending whether it's the old style or new style breech face. Each gun ... and particularly each individual slide when it comes to how the extractor spring hole was machined ... can make for some variance in older guns in the proper extractor spring tension needed for the gun to feed & extract properly.
