PAB, can you ask your management company what the Hungarian name for this tax is?
In relation to your second query, where you hold your Hungarian net rental income is not of primary importance. You can send it to Ireland if you'd prefer.
There is a double taxation treaty between Hungary and Ireland, which means that you can deduct the tax paid in Hungary from your Irish liability. Your Irish liability is not obvious from your post - e.g. you may be in the 41% tax bracket and would therefore be possibly liable to pay additional tax in Ireland. Your accountant will be able to advise on your specific situation, as deductions, etc. may be made on your Irish liability.