What does it mean if my cat's pink nose turned pale with black dots?

Maybe not, however a change in the "pallor", or color of the surfaces of the inside the mouth, the nose (in a light-colored cat) or sclera (white part) of the eyes can indicate certain disease processes. I'd worry potentially about anemia, or lack of red blood cells, making your cat's nose look pale-r than normal. Anemia can have a number of causes including certain infectious diseases transmitted by fleas as well as autoimmune disease. The black dots are concerning as well, although without seeing a picture it's hard to say what those are. Bottom line - I think this is worth getting checked out by your vet, to be sure your kitty is normal.