Hi Istvan, i think you must to incorporate a company in Cyprus and choose a bank like bankofcyprus.com or hellenicbank.com. However you must start by opening your company with a tax lawyer company like
Cyprus Company incorporation | Offshore | Shelf companies & Meridian Trust or
BDO and They can advise you a bank.
You can give as activity: Management of Internet websites
I think the best would be that you move on site !