Hello & welcome, I am Dr. B, a licensed veterinarian and I would like to help you today.
Now based on his signs, we'd be quite suspicious of cat flu (aka feline upper respiratory infection) here. Especially as both had signs initially. Though for him it sounds like he may also have a GI bug compromising his immune system or possible just has a weaker immune system generally compared to the other cat.
With this all in mind, as long as he doesn't have any snotty/green/yellow discharge from his eyes or nose (a sign of bacteria and a sign as antibiotics would be indicated), we can try some supportive care to ease his signs and help his immune system. To start, we can start with steam treatment. To do so, you can take him in the bathroom while you run a hot shower. The steam will help loosen and clear discharge from the airway. You can also use a baby nebulizer, but often they don’t like things held up to their faces. So, if you find that to be an issue, you can make a little ‘steam tent’ with him in a carrier, the nebulizer next to that, and a bed sheet over both.
Furthermore, if he is building up mucus that the steam isn't shifting, use a cotton ball moistened with warm water to wipe away crust and mucus. As well, non-medicated saline nasal drops (ie Ocean Mist or Little Noses) can be used. To do so, just tilt the head back and drop 2-3 drops in one nostril. Not a favorite, but it helps. After the drops go down, you can let the head up and wipe away any discharge that gets loosened. Then repeat with the other nostril.
For any coughing, you can try a mild cough suppressant. Often we'll give a few millilitres of plain honey & glycerin cough syrup or plain honey. As well, making sure he is getting food and water is important. For the nausea, you can try an OTC pet safe antacid [ie Pepcid (More Info/Dose @http://www.petplace.com/article/drug-library/library/over-the-counter/famotidine-pepcid), Tagamet (More Info/Dose Here @ http://www.petplace.com/article/drug-library/library/over-the-counter/cimetidine-hcl-tagamet)]. Whichever you choose, we’d give this 20 minutes before offering food to allow absorption. Of course, do double check with your vet if he has a known health issues or is on any medications you haven't mentioned. Though if he cannot keep it down due to nausea, that is usually a red flag that we need to bypass his mouth with injectable anti-vomiting medication from his vet. If that does settle his stomach but he is very congested, you could also try offering smelly wet foods (since they are high in water). It may help to warm it up a bit in the microwave to help him smell it. And OTC cat hairball gel can be added to these meals to push any bones/hair from those small animals through the gut if need be.
Furthermore, since feline herpes virus is a suspect, we can also start OTC L-lysine. This is a nutritional supplement that can help them recover quicker. This is available over the counter at vets, pet stores, and even online. They come as gels, powders, and as crushable tablets that can be mixed into food. An average cat dose is 500mg a day.
Overall, this does sound highly suspicious of an upper respiratory based infection which one cat is getting over but this lad is struggling with. Therefore, you can try the above to help give him some relief. Of course, if these signs linger or are severe or the vomiting continues despite the antacid option, then we'd want the local vet involved. They can confirm our concerns and dispense cat safe decongestants (since human ones are toxic for kitties), injectable anti-nausea medication (as they check for any wildlife induced gut blockages), +/- antibiotics to help us nip this in the bud.
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