Archer vs Rider
Editorial department (Prediction)
At long range his fire won't work against Bellerophon.
At medium range the bow is really handy, and Rider who's inferior in skill only has her Mystic Eyes.
But if Archer's bow starts to get petrified, he can activate Unlimited Blade Works.
In which case Rider will just use Bellerophon and destroy him instantly.
He can't shoot them down but Rho Aias can buy him time enough to pull out Harpe and maybe have a chance at victory.

Nasu-san CHECK!
You would think Archer would have the advantage, but Rider actually has the edge at long range? Just like that question, their compatibility is not very good. Since both aren't fixated on the methods of winning, they'd definitely both choose a low-risk and high-return type of warfare.
Even if Archer takes out a Saber-class holy sword, he doesn't have enough magical energy to draw on to maximize it, so it'd be hard to for him to attack Bellerophon. What determines victory or defeat is whether Aias on its own would be enough to block Bellerophon perhaps...?

Dengeki Hime 02/2006, Type-Moon Double Maniax - [Fate] Dream Battle - Archer vs Rider