Oh Sveta I am so glad to hear that your kitty is improving!! When cats get ill the situation usually deteriorates quickly (don't ask me how I know ), so very very glad you took him to the vet immediately. Eating is most definitely a good sign, even if it's just a little bit (when they're not eating you need to REALLY worry).
If you haven't already, switching Winnie's food over to a sensitive systems type could help, especially with vomiting. My cat is older and was vomiting quite often a couple years back, and our vet had me switch his food to a sensitive system type. It helped stop most of it, which in turn helped boost his stomach strength and health; seemed to be one of those "once he starts throwing up he'll keep throwing up" things, and also actually helped his liver as well.
Sending you and Winnie well wishes and snuggles!