There are five adults and a 5 yr old child needing to get to Valencia Airport for 4am 25th May and return 10am 1st June.
Is there a bus that goes either from Alicante Airport or Benidorm bus station?
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There are coaches from Benidorm to Valencia airport, run by Alsa, but not very many. You would have to leave Benidorm at 15.45 the day before, arriving at the airport at 17.45. On the way back, there is one that leaves just too early for you, at 09.45, arriving 12.00, and then one at 13.30, arriving 16.00. As far as I can see, you can buy an open return which would be valid on either one, and would cost a total of €220.75 for 5 adults and 1 child.
You could get a coach from Alicante airport (same website) but it seems to involve a change in Benidorm anyway.
The issue with this is that it does not run that early in the morning, meaning that you'd have to leave the day before. You should be fine on your way back.
Though you ask about public transport it seems there may not be much very convenient for a 04:00 arrival and therefore you may want to consider a taxi (minivan). It would be more expensive than public transport but could be far more convenient without being uneconomical, if it saves overnight accommodation charges for five adults and one child. The journey time should be under two hours so you would need to leave around 02:00.
Suntransfers.com offers the return service for successive Wednesdays (end November 2016) for 689.26€.