I have taken my grey on many trips in the car and I do not cover her, she likes to see where she is going. Since your grey is very timid and fearful you could try it and see how it goes, if it upsets him too much then covering may be the answer.
You really should take your grey on outings in the car to get him used to riding and going places and one place he probably should be going is to the vet, most of us take ours for a yearly checkup just to make sure everything is fine.