You can paint a bathroom sink, but it's going to be hard to make it look great. A basic sink is pretty cheap. Why not just replace?
In many areas there are companies that come out and spray it with epoxy.
XIM Tile Doc - its a spray. Works great!
Rust-Oleum makes a product to refinish countertops. Comes in different colors and styles, check Home Depot. It is a process, but worth it if you have the time.
Is this a metal enameled sink? or one of those "acrylic" molded counter top deals.
a few years back I replaced a basic metal sink in my guest bath, I picked up a new "matching" sink in enameled steel for about 25 buck. This sink here
is less than 40...and is porcelain
This is a square sink that is marbled porcelain looking kind and they have built the counter top around. I would have to replace the whole vanity and countertop and Im not wanting to do that
If the dimensions are pretty standard you might be able to get just a replacement top.
the trouble with painting sinks is the constant "exposure" to water, the basic counter top "refinish kits" work well on the counter because they are not exposed to water as much.
Commercial sink and tub refinishing use heavy duty "epoxy" type paints that hold up much better that your basic "spray" paint.
This is a bathroom I did that had the cultured marble countertop with integrated sink. I used the XIM Tile doc on the sink. I also faux finished the countertops to look like granite.
For the sink portion it is a product called xim tile doc; it's a spray. The countertop was done with xim primer and painted to look like granite then sealed with Aqua brand sealers. I'm not at home now but let me know if u want the sealer info and I can get it when I get back.
If you have a laminate countertop, you can paint it with a kit from RustOleum Countertop Transformations - rustoleumtransformations.com (kinda pricey) or try Gianai - gianigranit.com which is a multi-step finish that resembles a natural stone.