Of course, the famous Dolly was cloned from a somatic cell, moreover, Dolly herself had already died at the time of cloning. (If your cat dies before you start making a clone, store its tissues in liquid nitrogen). Now this cloning technology is patented and is called kernel transfer. It should cost a lot, but your cat will do a good deed by sponsoring science research.
Really. Cloning requires the whole genome of an organism. You mean a cat has no sperm due to castration, but a sperm has only half of the genome. You need to take any cell of the cat (geneticists themselves know which one is better), remove the nucleus from there and transplant it into the cat's egg, before that, the nucleolus must be removed from the cat's egg. And from this oocyte with a complete genome a cat will develop genetically identical to the original cat