Your fan looks worse than mine and that's saying something!
But yeah compressed air works fine if the dust hasn't settled there for a long time, I would suggest to keep a vacuum cleaner away from the actual parts since it can create static electricity which can potentially fry a motherboard, but unless you are
really careless with it and/or use some sort of brush tip at the end of it, you should still most likely be fine.
In any case, vacuum is still great for picking up all the dust, hair and anything else that may fly out.
But since it looks like the dust on yours has settled for awhile, compressed air itself will not do the whole job unless you go for crazy PSI that would be pretty hard to pin-point to the exact area you are trying to clean without professional equipment anyway.
So you may want to invest in some anti-static brushes, they pretty cheap on Amazon and you don't have to worry about frying anything by brushing with those, but the fan itself should be fine if you don't actually touch a circuit board anyway, so if you feeling extra cheap just use an old toothbrush and be extra careful not to rub on anything with circuitry.
The E-duster you linked to, or similar models are actually a great in-between, a full blown compressor and a compressed air-can, most suited for electronic use such as cleaning a PC.
There's actual PCB specific cleaners if you want to go all out, like
this one which is also pretty cheap, isopropyl alcohol of high purity can also work on a pinch.
If you haven't done it since you got the system, and unless you planning on upgrading it soon, you may want to replace the thermal paste on your CPU, which is a more complicated process, but there's many how-to's on YouTube to get you over the basics.
Just to to be clear, if you haven't done this before and are generally inexperienced with electronics, you may want to be extra careful and even invest in an anti-static mat/gloves/strap. Though usually people that fry their systems are either living in very dry climates (good luck trying to get static electricity going with 90% humidity), do something really careless, or are just plainly very unlucky, so I always say better safe than sorry, unless you don't care that much if you plan to replace/upgrade the system soon anyway.