There are several things that helped me with my anxiety. First and foremost, ask your doctor if there's medication that might help you. My oncologist gave me a prescription for diazepam. She said to take one before I left home, and another when I arrived at the cancer center. It definitely helped.
I also like to pretend. During stressful procedures I sometimes pretend that I'm scuba diving. I close my eyes and tell the doctor or nurse what I'm doing, and ask them not to talk with me unless they have to, so as to let my imagination sweep me away from the procedure.
If you have anyone in your life who you find cheerful and/or calm, ask them to come with you. Let them hold your hand, or look into your eyes, or whatever it will take to distract you.
You can also get a numbing agent for the skin over your port. Most people used it. There is also a numbing cream you can get from the oncologist. As I recall, they said to put that on some period of time before you go in. They can also give you a shot, although of course that's still a stick, though a smaller one.
In your case, perhaps prayer of some kind will help. Start well before you go in, with requests for calm and peace and courage. Ask, in your imagination, for a strong supporter. I sometimes imagine a lion walking beside me, lending me his strength and courage. Or imagine an eagle flying above you, or a gentle baby animal that you are protecting. The mind is a powerful and amazing thing. Let it work for you.
I will be in NC within the next week or so. Perhaps once I get settled in, and through opening weekend, we can get together to talk.