Will my 13 year old cat ever accept my 6 month old kitten

This is about pets, not about friends and family and I have asked that it be moved.

I'm a "dog person" and know little about cats; however, I had an older dog and a puppy and the dog was afraid of the puppy because the puppy was so active, kept jumping at him, wanted to wrestle and (in general) gave him no peace. I had to keep them apart unless the older dog wanted contact and, yes, they became friends and companions eventually.

Introducing a new cat into your household, while another cat lives there can take time. You should let them be around each other, and introduce them.
How long have they been together?
My vet gave me advice on this, so you can consult your vet.