There exists a fantastic resource for finding PhD/Postdoc positions across the EU which most of you will know (eureaxess). I was just wondering if anyone could point me to an equivalent resource which focusses solely on Asia (Japan, Korea, China ...)? Does such a site exist?

Not that I know of. One of the reasons being that there isn't quite the same level of international R&D collaboration in Asia as there is in Europe. So, I don't think there is an umbrella management scheme for R&D that exists within the EU.
I'd suggest checking the usual PhD sites like findaphd.com and phdjobs.com and specifying Asian countries in the search.