The roadworks will begin on Monday and remain in place until at least 28 February.
Although the longer diversion applies only to vehicles over 3.5 tonnes, councillor Huw Murphy, who represents the town on Pembrokeshire council, said businesses were worried about the effect of the work.
He said: “The problem for us is that in Newport there are no easy diversionary routes.
“The road will be closed but Newport won’t be closed. I do think it’s going to have an impact on the local economy.”
He added the town relied heavily on tourism, hospitality and leisure and there was a “real risk” that would “fall off”.
