I don't think there's a single answer here.

In most human interactions you start with really truly trying to listen and observe and understand.

Often when someone is scared or grieving you don't need to solve their problems or concerns, you just need to enable them to express their feelings. Likely everyone they've interacted with during their journey has been too busy or hasn't cared enough to listen to them.

When I'm absolutely miserable the simple act of crying is an emotional release and I invariably feel better after. Acceptance and Commitment Therapy suggests that it's the resistance to emotions that causes us harm - if you've spent the last several weeks trying not to worry about an operation it creates enormous tension - sometimes just being able to tell someone what you're worried about is a tremendous relief.

If someone shares medical concerns like "what if x happens during the operation" I wouldn't engage with that and simply say "that's probably a good question for the surgeon when he visits, but I'm sure he has a plan to manage that risk".

I'd also suggest asking the doctor for something to help them relax.