A cat has been a cat for all of recorded history.
By cat, I presume you mean the common domesticated cat, felis silvestris catus.

It speaks of cats on Noah's ark.
Which was, what? 4,000 years ago? It is absurd to expect a species to evolve beyond recognition in that space of time.

If no change appears in a cat for thousands of years, how does that prove evolution?
It doesn't, but neither would it disprove it. In any case, cats most certainly have changed. Have you considered all the different varieties of breeds of cat? They have not yet been selectively bred long enough to be incompatible with other breeds, but with a much longer amount of time, it would happen. The fact is, human have been on the earth for hardly an eyeblink.

Or, perhaps the cats will not significantly evolve. Some species, such as sharks and crocodiles, have not needed to, because they were adapted enough for their habitats, which didn't massively change. They did evolve a bit, mind you, but not to the extent that other groups, such as mammals, did.