This man was not your gg grandfather either. This is the first man on Michael's list, shown as Andreas Klein, age 21. Same company and regiment, but just spelled differently on the NPS web-site than it was in the New York records that Michael has provided. Spelling errors were common, especially when a clerk might be listening to someone with a foreign accent, or it could have been a transcription error made after the war.
I may have been in error about an exemption due to occupation. That seems intuitive, but I read through the Enrollment Act of 1863, and it seems that even those with critical job skills were not exempted. It may have been that he, or maybe even his employer, found a substitute or payed the $300 fee to avoid the draft. Another possibility is simply that his name was never drawn. Not all who registered were actually drafted.