Hmmm...not exactly sure if we know exactly when this happened, or even if there's a guesstimate for this. However, I would say to never trust a ghost for information about dates and times. They probably have a different perspective on that kind of thing.
"The early 20th century" is probably too early for him to have been active. I would hazard a guess that it was probably a much more recent thing, happening in the early 21st century, maybe right before the first Rikti invasion.
Edit -> If the Hamidon created the Devouring Earth, and the DE captured and killed the Woodsman, it must have been a more recent event. Dr. Hamidon only turned himself into a goo-monster fairly recently in the game world.
Let me never fall into the vulgar mistake of dreaming that I am persecuted whenever I am contradicted.
~Ralph Waldo Emerson
"I was just the one with the most unsolicited sombrero." - Traegus
Per Paragon Wiki's in-game timeline, Hamidon created the Will of the Earth and then took his current form in March 2003. Other DE creatures were sighted shortly after. So Woodsman would have died sometime between then and April 2004 (the game's release.) No idea about when he got started though.
http://paragonwiki.com/wiki/Canon_Timeline
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@Inquisitor
Thank you for providing at least part of the answer. I'm unsure how I missed that in the timeline, although it's probably because I didn't look far enough down the page.
I'm attempting to work out when the Woodsman was an active, breathing hero. Scout was looking for him and himself was known as being active in "the early twentieth century," but the wording does not rule out this mission being after that time frame. The Devouring Earth captured him and used his powers to convert Woodsvale to Eden, but I'm having trouble working out the date on that event either. The Terra arc implies the transformation is fairly recent, but the Numina TF suggests that the Woodsman has been missing for a considerable period.
TL;DR- When did the Woodsman's hero career start and stop?