On RENFE website, looks like first train departs at 7.18am and arrives in Alicante at 9.04am. Then 15-20 min taxi to airport........as tight as a duck's @rse.
What about a bus! Departs 2am, arrives 4.30am! Don't fancy it myself but needs must I suppose.
http://www.alsa.es/portal/site/Alsa/template.PAGE/menuitem.a2b8c42c4264a03c66c766c7421000e5/?javax.portlet.tpst=28b12525bd34c9fec845c845421000e5&javax.portlet.prp_28b12525bd34c9fec845c845421000e5=_spageview%3D%252FSearchExpeditions.do%26_kraview%3D1&javax.portlet.begCacheTok=com.vignette.cachetoken&javax.portlet.endCacheTok=com.vignette.cachetoken