Do fluctuations in radioactive isotope decay rates affect carbon dating techniques?
First off, the purported variations were reported to be cyclical. So the net effect on archeological dating would be null even if these variations existed.

Secondly, these purported variations are very dubious. Scientists are not immune from "crackpot syndrome". If anything, scientists are more susceptible to this than the average public. There's a bit of a crackpot lurking in every good scientist. Their job, their very being, is tied up in pushing the boundary of human knowledge. Very good scientists listen to their inner crackpot but keep it inside until they are very sure that their inner crackpot has revealed something new. Not so good scientists regularly expose their inner crackpot for the whole world to see.

What this means is that when you read some extraordinary claim you should be extraordinarily suspicious, at least at first. They might have found something, then again, they might just be exposing themselves.

In this case, I suspect the latter. What I suggest in the case of extraordinary claims: Go to a site such as scholar.google.com and search for the journal article in question. Scholar gives a citation count (oftentimes inflated, but that's a different story). Click on it and you'll get a new page that lists the articles that cited the article in question. In this case, you'll see that there have been a paltry 25 citations in the 8 years since the article was published. That alone is very noteworthy. This article that you think is significant is anything but.

If you dig deeper, you'll see that almost all of those 25 citations are self-citations, subsequent articles by one or more of the original authors who have cited their own work. That makes this article even more suspicious. If you filter out the self-citations, you'll see that the original list of 25 citations dwindles down to two or three, and most of those say "There's nothing to see here. Move along, move along."