"Sherman, set the Way-Back for Vicksburg, 1863..."
I actually think the progress that occurred in the decade of the 1860s was amazing: From single-shot muzzleloading percussion muskets to magazine-fed, mass-produced metallic cartridge rifled repeaters, etc., not to mention the progress in the span of one human lifetime to most of the firearms innovations that now exist.
I submit that even if you hopped into the Way-Back with Mr. Peabody, you would face insurmountable obstacles to making an impact on the outcome of the Civil War even with knowledge of advanced firearms technology. You'd have to come up with advanced brass, springs, metallurgy, manufacturing quality control, geez, jacketed bullets, powder, lubricating oil, etc., technologies from a relatively primitive start. Also, you'd be competing with every other stripe of kook trying to mobilize the political and financial support (there was no shortage of inventors peddling wonder weapons). It would take something like the Manhattan Project (well, not quite) and, I think, more than four years.
Not to be a wet blanket, but I think that history is a complicated tapestry, with technology belonging to a time. Even if you could transport, say, an Abrams Tank back in time, it would be a short-term thing because you could not sustain it with the logistical infrastructure of the time.
Perhaps the more compelling mental exercise is what could be done to optimize the existing technology of the time, like bringing the tactics and technology of, say, 1870 to 1860...