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 En?ma 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
well its different Gil's EA has only the power to wipe out 1000 people in one 1 epic blow which ATG's entire army is gone and then after being chained by endiku he got GOB to death meaning Rider's RM can only summon 999 servants max while archer has a unlimitied amount of blades and noble phantasms in his arsenal.
EA takes about 2.5 secs to get a full power charge while if archer summons excaulbur or black excailbur even if its down one rank it certainly can wipe out a part of his army since its out is like EA's except its weaker due to it being a copy and it isn't antiworld and is not able to wipe out 1000 people.
Gil cheated to win also Rider's sword that has no power like the likes of excaibur and EA yet was able to spar with them without much effort.