My cat's right eye is a little bit puffy, and is now pink all around it. Can cat's get pink eye? Is it something worse? Is there a home remedy or should I take her to the vet ASAP?
kitty had distemper vaccine 4 weeks ago

Yes, cats do get pink eye,.. conjunctivitis.
Bathe with saline water (cooled boiled water with salt) to soothe and get an anti-biotic ointment from vet.
You can use Polysporin Eye/Ear ointment OTC fr. drug store.
But if in doubt, take kitty to vet.
I've treated many cats with minor eye infection with cured results. But BE CERTAIN it pink eye infection and NOT a foreign object in eye, etc.
Gd. luck.

let's bring him to the veterinary clinic