An internet search turned up this info:

"The word sharpshooter stems from the Old English (before 830) word 'scharp' appearing in the book 'Ancrene Riwle.' It meant pretty much what it means today; cutting, keen, or sharp. It's a relative of the Old Frisian 'skerp' also meaning sharp. The modern Dutch word is 'scherp' the Old High German 'scarf'...modern German 'scharf'.

Our word 'sharp' was first attached to the word 'shooter' (sharpshooter) in 1802. It's meaning, of course, one who is keenly accurate with a gun."
http://en.allexperts.com/q/Etymology-Meaning-Words-1474/etymology-sharpshooter.htm

And:
"I found the term in the Edinburgh Advertiser for 23 June 1801, in an item on the North British Militia: "This Regiment has several Field Pieces, and two companies of Sharp Shooters, which are very necessary in the modern Stile of War." It quickly became common, appearing in the Times more than 20 times in the next three years. In 1805, a report could say baldly in the expectation of being
immediately understood that "Lord Nelson was wounded by a French Sharpshooter."
http://www.worldwidewords.org/backissues/wbi070407.txt

Then again:
"In fact, sharpshooter goes back in Germanic Europe at least as far back as the early 1700s or so, when the modern rifle-armed troops were first used in the Austrian and Prussian armies, and probably has a civilian origin. It survives today as the Germans have never adopted the term sniper for precision shooter and continue to use scharfshutzen instead."
http://brettschulte.net/ACWBlog/archives/guest_blogging/origin_of_sharpshooter.html

In any case, the word certainly predates the Sharps Rifle.

Geoff Walden