Instructions
Ask someone else to deliver the new cat to your home. When the new cat arrives, act like it's no big deal. The worst thing you can do is fawn all over the new cat. If you suddenly pay a lot of attention to a new pet, your old cat is likely to feel territorial and angry, so be casual and let the old cat take the lead.
Put the new cat into an isolated room and bring the carrier back out into the open. Let your old cat sniff the carrier and pick up the new cat's scent, but don't force the issue and don't act like you're paying a lot of attention to the process. Act as normally as possible with your at-home routine and allow your old cat to adjust with time.
Spend lots of time with your old cat. The new cat needs time to adjust to an unfamiliar environment and should get some time alone in the isolated room to explore. Take this time to shower attention on your old cat to reinforce the idea that your affections aren't going away just because of a new cat.
Introduce the old and new cats face-to-face, under supervision, when there is no hissing or hostility toward the new cat's carrier or under the door of the isolated room. Expect a little bit of aggression as the cats mark their boundaries, but prevent the cats from hurting each other. Return the new cat to the isolated room, with food, water and litter, when you are not around to supervise.
Reward good behavior, even if it seems small. When the cats get along well or interact without hostility, give them treats and attention to encourage them to get along. In time, they might never be best friends, but they can work out an acceptable situation where the new cat fits in without the old cat getting upset.
How to Bring a Second cat into the Family and Avoid Upsetting Your old cat
Bringing a second cat into your family can add love and companionship for both you and your old cat, but that doesn't mean your old cat might not get upset. With a little time and a lot of patience, you can make the transition easier for both cats.