Alcala is a fair way out and I think a taxi would cost at least 55€. The cheapest way (though hardest work) would be the metro to neuvos minesterios and then the cercania from there.

It would also probably be cheaper to pick up a hire car at the airport. If you don't need it afterwards, maybe Europcar have a place in Alcala too and you could do an A to B type rental. Check their web site.