Age of the universe vs. its contents A couple of decades ago I read an article (could have been either Astronomy, Space, or Science News) that suggested the contents of the universe may be older than the universe itself.  Of course, that result was so preposterous that it was summarily dismissed.  However, when I read that article, I thought it made sense.  Surely the universe is much more massive than its contents, so as General Relatively predicts, should experience time more slowly.  This conclusion seemed so obvious that a) a bunch of other folks much brighter than I would have thought of it, and b) therefore my conclusion is completely erroneous.  So my question is: why wouldn't the mass of the universe cause it to experience time at a rate slower than its contents as GR predicts?  (There are a couple of questions on SE that are similar, but I do not believe this is a duplicate.)
Models are uncertain

I'm guessing that you are referring to age estimates of very old stars. The age of a star can be determined from certain observables, among one is its absolute luminosity, which in turn requires a precise measurement of its distance. Even today, the uncertainties involved yield estimates with mean values longer than the age of the Universe, which of course is impossible. For instance, the age of the star HD 140283 was found as late as in 2013 to be $14.46\pm0.8$ billion years (Gyr). This is not really in conflict with the estimated age of the Universe, $13.798\pm0.037$ Gyr; all this tells us is that our methods are still not perfect, but that HD 140283 was formed shortly after the Big Bang.

Time dilation is small

General relativity predicts that time evolves more slowly in a graviational field. This is with respect to an observer outside the gravitational field, and since there is no external observer wrt. the Universe, the global time of the Universe can be said to be the "right" time. You're right that in principle the total mass of the Universe makes time run slower than in a hypothetical, empty universe, but almost anywhere in the Universe, the gravitational field is way too weak to cause any noticeable dilation. Close to massive objects, time does run slow. On the surface of a star, the time dilation factor is roughly $(1 - GM/rc^2)^{-1/2} = 1.000002$. Inside the star, the factor is even closer to unity, because only the mass between the center and you contributes, so the farther you dive into the star (kids, don't try this at home), the less the time dilation. Only if you compress the star, so as to squezze it into a neutron star of a black hole, will you get a larger time dilation. For instance, at the "surface" of a black hole, time stops (but again, only wrt. an external observer; for the person at the surface, time evolves as expected)