About less then 50% of article is recap from anime, my opinion of Rin's thoughts, but it has strong point in connection so read it all...

Connection of Archer and Rin:
Why Rin summoned Archer- Heroic spirit Emiya?
In order to summon a servant you need catalyst (something that he possessed in past, something precious).
Shiro was wounded by Lancer and was near death, hopefully Rin made it in time to heal his would with her precious jewel that was passed down in her family in generations. After she was done healing him she regrouped with Archer and they went home. She left that used up jewel, with no mana in it. When Shiro woke up he wondered how could he survive that stab in his heart. Then he noticed that jewel... he could remember that someone saved his life but he was almost unconscious so he only knows that that person used this jewel to save him... He took it with him as lucky charm. When Rin woke up next morning, in anime we could see that she found that same jewel in her bed... She was sure she left it. My opinion that Rin and Archer were having fun in bed, or Archer saw how Rin was looking sweet and lied beside her. *your choice on this* As story went you could see how Rin was bothered by something. She was suspicious about Archer... I mean he couldn't remember his name, she could never find in history books or so that some servant like that projected. So she wanted then to try out something. She moved in Shiro's house... Because he was lazy magician and they were in truce she wanted to make him more capable for fighting. So Saber trained him and Rin was teaching him how to use magic, as you all know. Because strengthening was his only magic he was capable of she told him to practice projection. It was tough for him, but in the fight with Berserker he projected Saber's lost sword Caliburn... Rin was amazed how he could project something like that, let alone survive after using that twice? That projection was perfect copy... so she thought that he is maybe.....
As the story further went the fight with Caster appeared... Shiro for that fight projected Kanshou and Bakuya. U can see expression on Rin's face when she saw that Shiro 100% perfectly, without any difficulties, projected them. So then she probably thought, Shiro is Archer.

Things that has relation with Shiro being Archer:
*Shiro was in past part of Archery club.
*His fighting style was the same as Archers. The problem is that he hasn't saw Archer fighting, only gap of fight with Lancer.
*Shiro kept that jewel as his most precious item and kept it with him his whole life... That is why I explained about catalyst. The catalyst this time is that jewel... Shiro kept him with him even in his darkest times of his life, when he became Hero of Justice. Rin was only using it as backup if she would lack mana so the summoning would be perfect, that is why she could summon only Heroic Spirit Emiya.
*When Archer saw Rin maybe he knew all the time who he was, just he didn't want to tell Rin because this was time when there is another him Shiro. He wasn't happy that he became Hero of Justice... he was agree at himself that he was fool thinking that he could save thousands of lives. He was going berserk, killing thousand people to save one... in the end he died in running ahead his ideal. So good thing happened to him. He became Heroic Spirit and he was summoned in the same time where his young self was living. So he made his mind up, that he will try doing anything to stop present Emiya Shiro becoming Hero of Justice. That is why he was arrogant and negative about Shiros ideal and his way of thinking. He was having doubts in whole anime if he should just kill himself, but he was wounded by Saber and was weak so he didn't try doing something stupid that could end up getting himself killed by Saber, so he in truce remained calm and watched Shiro from distance and tried to make him realise throw words. That is why he when he saw Shiro how he cared about Saber's life only, his beloved girl, but mostly because he wanted that his dear Rin lives. So he couldn't care less by being bait. His wish was only to kill himself, but he had good feeling that he wont become like himself. Shiro dropped his ideal and was only saving those that could be saved... In the end when Shiro said that he will always remember in his heart that he loved person named Saber. In that key moment we could see that he dropped his ideal for his loved one, that he will pursuit her till the end of the world just to be with her, even that he knew that isn't possible.


Archer is Shirou who came from a parallel universe and from the future. He lived his life almost like Shirou did but unfortunately he never got close to Saber, Rin or Sakura.
He continued to believe his believings and that eventually saved many by killing many and thus when he realised it he became bitter. Also the people he saved eventually betrayed him and that lead to his destruction. He also was kind of underdog of the people who started a war and he was between and he was jailed.
Sometime in the middle of this he made a pact with the world and gained power, thus he became a 'heroic spirit' and was able to be summoned. (Actually he became a counter-guardian... But that is another matter for now.)

So he was betrayed, jailed and his believes killed him eventually and then he wished that if there would be a 0.1% chance that he could make a future that is not like his then he would sacrifice everything to make this future.

And he wasn't shocked that he met Shiro since he wished to meet him eventually but he was shocked that he saw Saber and that's why he was unable to block her first attack. And Saber eventually thought he was mocking her knightship.