Who exactly is Archer from the Unlimited Blade Works movie?
The Archer-class Servant in the fifth Fuyuki Grail War is: 


   Shirou Emiya from an alternate timeline than that of the Fate/stay night timeline. The Shirou from the alternate timeline made a contract with the world and became the Heroic Spirit EMIYA.  
 He is not a proper Heroic Spirit in the sense of other heroes (e.g. Saber, Lancer, Berserker). He acts as a Counter Guardian for the world as payment for the contract. 
 He is summoned because he still has the jewel pendant that Rin used to save his life after Lancer stabbed his heart. The jewel he owns is the actual catalyst used for his summoning rather than the one that Rin still has at the time of his summoning.