Vomited three times since around 9:50 and had one bout of diarrhea

If a cat is both vomiting and has diarrhea, it may have gastroenteritis or a viral infection such as feline panleukopenia. It is recommended to take the cat to a veterinary hospital for panleukopenia and other infectious-disease testing, plus a fecal exam for parasites, and to treat based on the vet’s diagnosis.

If hospital fluids and anti-vomiting, anti-diarrhea, and anti-inflammatory injections are not an option and the cat is otherwise stable, you can try symptomatic treatment for gastroenteritis: use an anti-vomiting medication, an anti-diarrhea medication, and oral antibiotics to prevent gut infection.

Once the cat recovers, give it active probiotics when possible to maintain gut health and prevent recurrence of digestive problems.