I got it to work when I removed $fields_string altogether. Instead I pasted the POST I got from firebug (in Firefox), which you can see after you do a search on the ryanair website.
curl_setopt($ch, CURLOPT_POSTFIELDS,'__EVENTTARGET=SearchInput%24ButtonSubmit&__EVENTARGUMENT=&__VIEWSTATE=%2FwEPDwUBMGRkwiC%2BdiO7YHrN9X8OU5GYY7FHcEU%3D&formaction=Search.aspx&errorlist=&SearchInput%24IsFlexible=on&SearchInput%24TripType=RoundTrip&SearchInput%24Orig=NYO&SearchInput%24Dest=WMI&SearchInput%24DeptDate=17%2F05%2F2014&SearchInput%24RetDate=22%2F05%2F2014&SearchInput%24PaxTypeADT=1&SearchInput%24PaxTypeCHD=0&SearchInput%24PaxTypeINFANT=0');