Rider will win against Archer.
Eventhough Rider is nothing against Gilgamesh, it doesn't mean that he can't defeat Archer.
Reasons why Rider lost to Gilgamesh:
1) Gilgamesh has a wide area of attack, his Enûma Elish is Anti-World, which is able to clear Rider's army in a split second.
1) He charges against Gilgamesh emptied handed, it's impossible to defeat the King of Heroes without a weapon
Reasons why Rider > Archer:
1) Archer uses anti-unit Noble Phantasm, it's almost impossible to clear Rider's troops of armies.
2) Rider > Archer in most if not all of the status
3) HE IS ALEXANDER THE GREAT