How does slow anti-hydrogen annihilate with normal matter in the lab? In a recent article:
Physical Review A 83, 032903 (2011), A.Yu. Voronin, P.Froelich, V.V. Nesvizhevsky, Antihydrogen Gravitational Quantum States 
the authors claim that anti-hydrogen has a lifetime of 0.1 seconds when placed in a container with metal walls. They say that the atom is kept from annihilating with the walls because of the Casimir-Polder effect. This raises in my mind the question of how it is that the atoms annihilate. My first guess is that the positron would annihilate against an electron, and then the proton would be attracted into the metal by the image charge effect. But this would seem to be in contradiction to the Casimir-Polder effect mentioned in the paper.
So how does slow anti-hydrogen decay in the presence of normal matter?
 A: Dear Carl, the correct paper to derive the 0.1-second lifetime of the anti-Hydrogen atom in the gravitational field is described after the very sentence you quoted.
There is a "[20]" symbol which means that the sentence is justified in the reference number 20 in the list of literature at the end of the paper you quoted. So the correct paper that answers your question is

Quantum reflection of ultracold antihydrogen from a solid surface
  A Yu Voronin and P Froelich 2005 J. Phys. B: At. Mol. Opt. Phys. 38 L301, doi: 10.1088/0953-4075/38/18/L02
http://iopscience.iop.org/0953-4075/38/18/L02
http://iopscience.iop.org/0953-4075/38/18/L02/pdf/0953-4075_38_18_L02.pdf

which is fully available online - click the last link for the PDF file. At distances longer than 15 nanometers from the metal, the Casimir-Polder potential has the form $-74/z^4$ in atomic units. At shorter distances between 1 Bohr radius or so to 15 nanometers, the potential becomes $-0.25/z^3$ in atomic units - a van der Waals form.
Only at distances shorter than 1 Bohr radius or so, the anti-Hydrogen behaves differently than the Hydrogen, and is annihilated. This portion of the potential is not probed in the 0.1-second case: note that to get annihilation, the positron must approach the electron at a shorter distance than the Bohr radius - comparable to the Compton wavelength of the electron (and even shorter, nuclear distances if we want to annihilate the hadrons). The authors calculate the reflection probability as a function of the kinetic energy of the anti-atom. The lower energy/temperature they have, the more likely they will bounce off. For low enough kinetic energy, the reflection probability is as high as 0.99987 (a table).
In those cases, one doesn't probe the short-distance portion of the potential. So the answer to your question why there's no annihilation is the same as the answer to the question why you don't get fusion if you fill a tank with the Hydrogen gas.
A: Let me say that antihydrogen is never " slow" in the process of annihilation, when it comes to matter it is already accelerated by Casimir-Polder potential to the energies of few eV, which corresponds to temperature of dozens of thousands of degrees. The attraction forces between particles and antiparticles tear antihydrogen into antiproton, which is captured by matter atom, and positron, which most likely forms pair with electron. Antiproton falls down on nuclei, ejecting electrons from the atom. Then it
Usually destroys nuclei and annihilates. This process usually takes time of order of 10^-9 s or so, electron-positron in best case could survive for 10^-7 s. 
