Fairy Castle Cactus is what I am told this cactus is. It was gift. Everything was fine until I left it outside for a few days. Then this brown spot showed up. I was told to cut the brown off. I did. Now it's mushy. The rest of the cactus does not have this. It's only this one part of the cactus. I was told to cut the brown part off. I did. But it still mushy. First picture is what the sick part look like. Second picture is what the plant looked like when I first planted in November. The last picture is what it look like now. Am I killing this cactus?

First, I think your plant has been mis-identified. It looks like a Euphorbia, not a Fairy Castle Cactus. When you cut it, did it leak white sap?

When you cut the brown part off, did you cut enough that the cut edge was completely greenish whitish all the way across? If there was brown down the middle, you didn't cut far enough. The problem is, whatever caused the rot could be from the roots or it could be the tip.

I reached out to the person who gave it to me. She said it was that.
When I cut, there was no white. Just green. The root didn't show rot of any kind. Should I cut it lower or just cut that branch completely

So it must be the Florida native (Acanthocereus tetragonus) but not the montrose version (Acanthocereus tetragonus 'Fairy Castle').

Cut the rotting stem lower until there is no sign of brown anywhere in the stem. Clean you knife in between each cut with alcohol so you don't spread the rot.

Okay. I will cut it. Hope for the best. Thank you.