Hi there and thanks for your question. It sounds like your cat is really not feeling well. It is hard for a cat to tell us when they are unwell, so lethargy and vomiting and inappetance are signs that he needs help.
Unfortunately it is often very hard to tell what is wrong with a cat as there are so many things that can cause the same symptoms. The list of possibilities is quite long. It is possible that he has an upset stomach from a change in food or from eating a new type of treat. It is possibles he has eaten a foreign object that is blocking his stomach. He could have a kidney or liver disease. Does she chew on any houseplants? Some are quite toxic. He could have a fever which is most likely if he was in a fight with another cat.
It is possible that he is fighting a respiratory virus. Usually though the congestion in the nose is really obvious as most cats will be very stuffy and sniffy.
It really sounds like your little boy needs a vet visit. If cats go more than a few days without eating they can have problems with their liver that are quite serious. Your vet will do a good thorough exam and should be able to tell if this is a respiratory virus, an infection, a kidney disease or something else.
If you can't get to the vet right away (but hopefully you can) you can try feeding him some canned food that has been heated up for a few seconds in the microwave to improve the odour.
I hope this information helps! If you have more concerns, just hit reply. Or, if my answer is sufficient, then please hit accept. Thanks!