There is also a NATS minibus service from Southampton Airport Parkway station (right next to the Travel Inn and on the same side of the railway tracks) which goes to CTC regularly during the morning and back in the afternoon. It is operated by Airlynx (blue with yellow writing).
I'm not sure whether you need to show NATS ID to board it. If you are planning on using it, I suggest you call reception at the CTC and explain you are coming for an interview and, if they need to, can they send you a letter or some form of proof that you can use it.
Phone no: 01489 616001.