Mia, you may want to ask the radiology place about their equipment or procedures. I think I had one as part of my radiation planning, but it was just an open tube. I barely remember, because I am not 100% sure it was the CT, but I think it was...and I was never closed in. There was no IV or contrast.
Last year I had a chest CT with contrast. They gave me an IV with a dye. I think I just laid on one of those narrow table things, and the machine moved and whirred around me. again, it was not closed in at all.
It should be a very quick process too, I bet the CT is over in a matter of minutes. But I do not think it is at all like an MRI, but I have not had an MRI myself.