I am not sure whether you are eligible but it's very unlikely to be automatic. In my experience, to get any type of refund, you have to submit a tax return (aangifte doen), if only to confirm that you haven't had any other sources of income than those the tax office knows about. Incidentally, you can still do that up to five years afterwards and they even sometimes send letters saying “you might be eligible for a refund” in year + 2 (I received one once).
If you would have stayed in the Netherlands without a job for the rest of the year, it would be reasonably easy: Download the yearly tax return software, click your way through and submit the data electronically. Here is the link for 2013 (unfortunately all in Dutch, the English part of the website seems to be exclusively about non-residents). You can do it starting in March of the following year and would receive a letter and a refund some time in summer IIRC.
Last time I checked (which is admittedly a few years back), the software was only for (relatively) simple cases and you were not allowed to use it if you resided abroad for any part of the year. I seem to recall than a friend did it anyway and I think she did receive a refund and was never bothered (as far as I know) for having used the software. Still, the proper way is to submit a different (paper) form (I have another friend who tried it, don't remember the details or how it worked out for him). Claiming some tax rebate while failing to properly account for your moving or your Irish income could even amount to fraud for all I know so be careful here.
Also, there have been some last-minute changes in 2014 that the tax office was not able to implement right away so that you will probably receive less money than you would have in other years or perhaps be asked to submit a tax return even if you did not have to do it for 2013. And I don't know how any of this interacts with the 30% ruling. In any case, do make sure the tax office has your new address, they can be nasty about it.