When I first sat down to investigate "best girl", I was pretty sure that this was a fairly old lexical item - not pre-internet, by any means, but probably c. 2003-04. Alas, I had fallen victim to internet time dilation yet again.
Having looked into it, I now claim that the idea of "best girl" as we now conceive of it only started to come into being quite a bit more recently - I would generously estimate c. 2010, or maybe a little bit earlier.
Of course, the idea of a "girl" being the "best" is as old as time. What we're really interested in is the idea of "best girl" as a distinct lexical item; as part of a particular grammatically-unusual construction, in which "best girl" is not accompanied by the definite article, often in a clipped utterance; as something that can be snowcloned (as it has been, to give us "worst girl" and "best boy"); as something so argot-y that it provokes confusion from people not familiar with the jargon of the internet anime subculture. In other words we're talking about things that are archetypically like "Ritsu is best girl".
I decided to take a look through the /a/ archiver - it only has posts from Feb 2008 forward (and I think its coverage is a bit spotty for 2008), but that seems to be good enough for our purposes. Why? Well, look at the search results for "is best girl" - there are about a hundred results between Feb 2008 and Jul 2010 (and thus almost certainly very few before Feb 2008), but a further 37,000 between Jul 2010 and now (Feb 2015). Some of that is certainly attributable to an increase in users on /a/ (see the bottom graph here for an idea of how much growth we're talking about - roughly a factor of three from trough to peak), but I think that far more of it has to do with the "meme" of "best girl" taking hold. (Implicit here is my claim that the idea of "best girl" as we now conceive of it is directly correlated to the usage of "best girl" without a definite article.)
Another good proxy for the memeticization of "best girl" is the growing use of "best girl thread" on /a/. Here, we see 25 posts from Feb 2008 to Apr 2011, and then a further 2800 from Apr 2011 to now (Feb 2015). Here, I think we can identify a fairly discrete changeover around mid-2011, when the phrase "best girl thread" began to sharply rise in popularity.
Beside that, where did the expression (or meme, I'm not sure it is considered a meme) originate from?
I am fairly confident that we can't pinpoint a specific origin for the term - as I mentioned above, the idea of a "girl" being the "best" is obvious and needs no explanation. The transition from discussion of "girls who are the best" to discussion of "best girls" as a crystallized notion was almost surely slow and gradual.
I should add: while I don't think we can pinpoint an origin for the term, I'm pretty sure that it was popularized on /a/ before spreading to the wider internet in the past two years or so. As the classic Human Centipede-inspired graphic suggests, a large fraction of anime-related memetic content (that isn't ganked directly from Japan) does come from 4chan.
Is this an expression that is used in Japan too, or is it a Western thing?
I'm by no means a 2ch memelord, but I don't know of any direct equivalent in Japan. Even if there is one, English "best girl" is assuredly an independent invention, and isn't inspired by the Japanese equivalent.