An anti-matter bomb has all the problems of a regular thermo-nuclear weapon, plus at least one other. First of all, the fall-out, whether from the bomb itself or the Martian rock it is detonated in, is fall-out - radioactive. Anti-matter doesn't have any magical properties, it's just a way to liberate the energy in (anti)matter a lot (orders of magnitude) more efficiently. Such a bomb would release a lot of energy, but the resultant fracturing of Mars causes us more grief than it's worth.
"At least one other" problem with an anti-matter weapon is providing sufficient anti-matter to build a weapon. Anti-matter isn't lying around - it long ago detonated itself, if indeed it was actually created (actually, just storing anti-matter in a matter world is another problem - it can not touch anything, else bang! whether you're ready or not). One of the issues with the Big Bang theories is "where is the anti-matter?"
We know anti-matter is real - we've made it. In vanishingly small quantities. There is a lot of energy in a chunk of matter, and the same amount in equal masses of anti-matter. Einstein's E = mc[super]2[/super] applies to both. That means that to create anti-matter, you have to supply the necessary energy in some kind of accelerator. I'm not sure what the world's total production of anti-matter to date is (not much - femtograms, if we really got ambitious), but the energy that all of it together could liberate is probably well south of a fly's fart.
Bombs just aren't the way to go - we can't build one big enough to supply the energy needed, and bombs apply the energy the wrong way - too concentrated, and too fast. To melt the core, you want heat, not shock, applied at a rate the rock can actually melt at.
Building a diamond cutter head for Apophis, even if we could, would be a waste of time. The diamond would just fracture (what didn't just vapourize) on the shock of impact (how do you think jewellers rough-shape cut diamonds? Same principle, and effective even on a vastly smaller energy scale).
I'd have to do some digging regarding relative masses/momentums/kinetic energies, but off-hand, substituting one rock (Apophis) for another (Phobos) isn't going to help. Still not enough available energy, and still largely shock energy, with only a side-order of melt on the Martian surface. Plus, Phobos is already handily located in the immediate Martian neighbourhood, and Apophis isn't, so there's a lot of additional energy expenditure just moving a rock to the target's area.
The only way you could possibly get any rock to orbit inside the volume of Mars is dig a cirum-Martian tunnel and evacuate all the air out of it. Given the right velocity (and the occaisonal course correction nudges), your rock could fall forever, but the object of the exercise was to melt the Martian core, not dig holes. I think this tunnelling is your reference to the 42-minute free-fall, but I'm not sure. Given that rock won't orbit through rock, I don't see the point of this idea. If you are looking for tidal heating caused by an orbiting body, sure, a smaller mass closer in will eventually achieve the same effect, all else permitting, but a tunnel-size rock is microscopic, gravitationally. Mars would radiate the tidal heating away as fast as such a small rock could cause it.
Olympus Mons and all of the other major Martian volcanoes are unlikely to be the result of plate tectonics - they are just too big to form near a subduction zone. Their massive size requires time to accumulate, and plate tectonics implies that the fracture zones move around, or seal up (plates can join - India and Asia, as an example). The experts can say better than I, but I don't think Mars ever had plate tectonics, and even if it did, it was certainly not for a long time.
These volcanoes are most probably situated over Martian "hot spots", weak places in the crust where plumes of magma could penetrate and reach the surface. On Earth, Hawaii, the Galapagos Islands, Yellowstone, and several other hot spots are known to exist (still active), even it their creation and behaviour isn't completely understood (what started the underlying plume?). Note that plate tectonics do not prevent hot spots: they just distribute the mass of the resultant volcanoes in chains of mountains - Hawaii, for example, is only the most recent volcano of the chain, and its replacement has already been detected forming under-sea. Hawaii is the biggest mountain on Earth (measured from the sea floor, not sea level), but it is smaller than Olympus Mons. If you added together the masses of all the volcanoes in the Hawaiian chain, then Olympus Mons doesn't do so well. It is bigger than any of the Earth volcanoes because there is no plate tectonics pulling it away from the hot spot. The Hawaiian total is greater because the Earth is bigger (more magma to vent), and the greater mass allows the plume to last longer.