Now I know this probably is well trodden territory, but this question has bugged me for some time, and I couldn't find a similar question in the archives (although plenty about relativistic time dilation and the twin paradox, which is closely related but not quite the same). The theory of special relativity suggests that two objects moving relative to each other (at near light speeds) will observe the other as moving at a slower time. However, both of these reference frames are local reference frames - they are not a global perspective that the universe is. So I'll give a little background before asking about the scenario, to show the difference between this and previous questions seen here (although I doubt I have been exhaustive in my search here).
At the time of Einstein, no one knew a means of measuring speed relative to the universe that was applicable everywhere in the universe (short of Mach's principle which Einstein supplanted in GR). Neither did they know the universe had a finite age, let alone a means of measuring it. But now we have both - we can measure the age of the universe by the wavelength of the MBR, and we can "in principle" measure the rate of change of the universe by the slow change in the MBR's wavelength. (If we really want to, we can also measure historic elements such as the rate and duration of supernova and the expansion of the universe.) Additionally our velocity relative to the universe can be known by measuring the variation in the MBR due to the Doppler Effect (assuming the MBR is more or less isotropic). That is, in the direction of travel, the MBR would be blue-shifted, while behind, it would be red-shifted, and the speed can be deduced by the magnitude of the wavelength shift. So we now have "in principle" methods of measuring both our velocity with respect to the universe and a form of universal time.
So if an alien is traveling at relativistic speeds to the universe, and measuring the MBR and it's wavelength change with respect to the internal clock of his ship, will it observe the universe to be aging faster or slower than we would observe?