Many Doctor Who fans have commented on how old William Hartnell looked at 55 compared to Peter Capaldi at the same age. There is a reason for that:
Hartnell wore a wig for the part, and also acted impressively well to portray a character who was supposed to be quite ancient, and not merely elderly. But he was also in the early stages of multiple sclerosis, which eventually forced him to relinquish the role, and that may also have been a factor in viewers getting an impression of more advanced age than was in fact the case.
Via Reddit, I also found this picture of all the actors who played the Doctor, when they were significantly younger than when they took over the role, but also indicating the age at which they did so:
The take-away message is that the Twelfth Doctor will obviously look older than his immediate predecessor, but will only look like he is the same age as the First Doctor appeared to be if he is made up to do so, as Hartnell was.