Funny no one is worried about the older cat giving the kitten anything.
A couple of years ago I wrote a bit about transitioning a cat/kitten to it's new home. Maybe some of it may help.
It is imperative you keep the kitten separated from other pets in your home until it has been health checked for all of their sakes. Kitten should have been dewormed at 6 to 8 weeks and again two weeks later before going to it's new home. It also should have had it's first shots. I love the fact that you will be taking the kitten to the vets immediately. Just try not to cause too much undo stress. Sometimes hard to do when kitten is leaving the only home it's known in it's short life. While at the vet please talk about the proper time to spay/neuter the kitten.
After coming home from the vet please ensure you do proper intros with the cats. A slow, gradual introduction doing smells only for the first few days and then slowly allowing sight, contact, etc., will make for success I'm sure.