That's the beauty of it: "at a time" can be arbitrarily long in real time. [1]
You don't even need to simulate every particle in the Universe -- if the simulation is run for you, you only need to simulate enough to convince your brain. Like when you dream and are convinced something's real, yet upon waking, you realize your criteria were extremely lax. For all we know, what we think of as being awake is actually a very hazy simulation presented to our "real" brains, which are actually much sharper and have been induced into a less-discerning state.
Something very interesting at Body Worlds 3 here at OMSI was the brain slices, which showed (according to the placards) that the part we "think with" is just the edges, and the interior is interconnections (grey matter vs. white matter). Compare the darker grey around the folds to the lighter grey filling up the rest:
http://www.med.harvard.edu/AANLIB/cases/caseB/054t_2.gif
So the actual storage needed for a brain's total memory may fit onto a hard drive today, at current estimates (possibly less than a terabyte). Working memory is much, much less. Communication (the interconnections) can take as long as needed.
[1] Another "cheat" would be to put the simulator on a spacecraft travelling at relativistic speeds in a tight orbit around a processing unit; when the simulator sends the processor packets of data to crunch, time dilation would give it extra CPU time.
Impractical to do on current computers - we'd need at the very least quantum computers (I'm guessing that's what the article means by "Some computer experts have projected, based on trends in processing power, that we will have such a computer by the middle of this century").
But you still run into physical limitations. Could one atom in a quantum computer be used to simulate more than one "virtual" atom at a time? More than O(n) virtual atoms? Do you run out of atoms eventually? Can you actually get infinite recursion that way? I certainly don't know, and I'm not sure whether the best quantum computing researchers can answer today.
Failing that, we'd need something more advanced than quantum computers - so really, at that point we're simply guessing.
Very fun philosophical subject, but I wouldn't go as far as saying it's "almost a mathematical certainty" yet, given our current knowledge...