Let me first say that it will be impossible to reproduce the ‘exact color’ of the original in most cases. There are likely colors that the printer cannot reproduce. And without really expensive capture equipment, the best you can hope for is acceptable.
Good lighting is usually the first step in art reproduction with a camera. The standard setup being two identical lights at a little less than 45º to the work, equidistant from the work and each other, feathered so that the left light is aimed at the right edge of the work, and the right light aimed at the left edge. You would meter the lights at the surface of the work and adjust so that there is less than a 1/10 stop difference between the center and corners.
If you don’t have an incident meter, you can shoot a plain sheet in place of the artwork, bring it into your editing software, and examine it for even coverage. Shooting tethered to a computer is enormously helpful.
As far as calibration is concerned, the monitor is the most important, and you would want a monitor with as wide a gamut as you can afford.
Printer calibration isn’t the same as the calibration of most other devices. You need a profile for the printer and the paper you are going to print on. And as already pointed out, many paper manufacture’s profiles are very good.
And no, you do not need to use the same device and software to calibrate your monitor, produce paper profiles for your printer, and calibrate your camera.
With a decent lighting setup, a calibrated monitor, you can always shoot a standard color patch card, something like a ColorChecker mini, on every other shot, and use that to manual adjust images. There are fairly simple methods of producing camera profiles: such as the X-rite ColorChecker Passport.