Dunno if ginger comes from the spice, or the supposed feisty nature of red-heads, esp. Irish. I think both have contributed. The earliest quotes (1820's-1830's) refer to horses.

The beast was ginger to the backbone- 1838 (There is earlier mention of treating a horse with ginger to stimulate him).
A ginger is a showy, fast horse- 1859.
Referring to people, 1840's. Talk Yankee to him and get his ginger up. (Get his dander up, 1830's).
Whether they were gingered up by the articles in the 'Times' or not I can't say- Disraeli, 1849.
I'll take the ginger out of him in short order- 1891.
He said that the umpires would not let him get out on the coaching lines and 'pump ginger' into his players- 1907.
I'll scatter a little ginger around all right- 1919.
A redhead...imbued with the quality that used to be called 'ginger'- 1945.
These quotes all from J. E. Lighter, "Historical Dictionary of American Slang," vol. 1.