Hints:
Don't use a vacuum cleaner, as SinxarKnights advises.
And the same goes for garage type air compressors or air lines, again, as advised by SinxarKnights.
If you're asthmatic, suffer from hayfever or have a dust allergy do this job outside, for obvious reasons.
A cloth is fine for use on the case but keep it well away from the electronic parts, it's very easy to get it trapped on one of those tiny, tiny surface mounted parts that litter the motherboard and very difficult to get it free without damage.
The pump should work like a charm, just be careful to keep the nozzle a safe distance from the parts you're cleaning.
A long bristled 1/4" paintbrush is a better tool than a toothbrush, it's gentler and the long bristles keep the hard handle and metal ferrule well away from your hardware. And use a natural bristle brush, nylon/rayon or other polymer bristles can in the extreme build up a static charge.
That paintbrush is also an ace tool for getting into the fins of CPU and GPU coolers, although you'll probably need to remove the cooler fans and cover to get right into the heatsink fins of a graphics card.
Never, EVER clean a CPU or GPU cooler fan with anything other than a cotton bud and even then be gentle, the blades can be extremely fragile, use just finger and thumb pressure and if required, get the bud wet to help it clean off more stubborn deposits.
Static is a vastly overrated threat, all you need to do is touch something externally grounded and you'll be fine: A masonry wall, a water pipe, a heating/AC duct, a gas pipe or a grounded metal cased electrical appliance that is plugged in will all suffice to keep static at bay. Assuming you're not doing this while: Stroking a cat with an amber rod, using a Vandergraff Generator, sliding up and down on a Nylon carpet or rubbing a balloon against a woollen jumper.