I have a black colored copper sink and thanks to lye that I poured to fix a slow drain, there has been a discoloration .
How do I paint it back again? 
How to paint a copper sink
coppersink
Related Solutions
What size is that copper? You can get plastic p-trap kits in 1-1/2" and 1-1/4", and these can be easily connected to your copper via slip ring compression fittings. Basically they consist of a nut with a cone-shaped plastic washer, which when compressed by the nut squeezes against the (copper) pipe to create a seal. Usually these are only on the sink side of things and the wall side has glued-on plastic but you can use the compression fittings on both ends.
Here is a 1-1/2" Plastic p-trap
If you want to solder, a better solution than using a compression ring on the wall side would be to sweat on a copper x female adapter, followed by a male PVC adapter and a PVC trap adapter. Better yet if you can find one, just use a brass copper trap adapter like the following:
A less desirable but acceptable solution would be to use a fernco fitting to go from the copper to PVC, and then you can avoid soldering.
It it is a new build, then removing the trap under the sink will not be an issue. Just loosen the nuts, remove the trap (have a pan underneath) and then clean the trap. This is most likely the source of the slow draining. If the trap looks clean, then get a hand auger (snake) to clean the drain line. It is best to do this with the trap out. Hand augers are typically half the cost of a plumber call. If it is still slow, then your drain vent might be clogged. At this point, it is usually cost effective to call a plumber to clear the line.
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Best Answer
Your sink probably isn't painted. Copper goes through several stages of oxidation (roughly the equivalent of iron rust). The final patina is the dark, durable finish you see there. Painting it would've completely defeated the point of building it with copper in the first place.
Your sink probably doesn't ever get to the green that article mentions due to the regular wear of use and cleaning. You'll want to just give it time to re-oxidize. There may be chemical methods of accelerating the process, but you risk doing more harm than good.