Bow & Arrow.
Xiphos vs Ild. Edge = Ild.
From the Mongol's horse, he can take big swings, and even if the Spartan has the armor, Mongol's got speed, and he will kill the Spartan.
Spear vs Glaive. Edge = Spear.
The Spear will definitely penetrate the Mongol's armor, where the Glaive, mainly a slashing weapon, won't.
Javelin vs Bow & Arrow. Edge = Bow & Arrow.
The accuracy, speed, and range of the Bow dominates the game, when the Javelin is slow and not accurate.
Shield vs Mace. Edge = Even.
Both are great crushing weapons, so I would have to make this an even fight.
A Spartan walks to the top of a mountain. As soon as he does, he attracts the attention of a Mongol on a horse a hundred yards away. He shoots his bow, and the arrow planted in the Spartan's shield. He jumped down, and ran closer, until he was fifty yards away. He heaved his javelin as hard as he can, but it missed the Mongol by a few feet, riding closer.
The Mongol shot another arrow, again hitting his shield. The Spartan threw the two arrows to the ground as an insult. He picked up his spear and ran to the horse, and jabbed it in his horse, until most of the shaft disappeared in it. The Mongol fell down, hard, and he dropped his bow. He picked up his ild, and swung several times, but each time bounced off the Spartan's shield.
The Spartan took the spear out of the horse, and jabbed at the Mongol, but he dodged it, and and sliced again. The sword cut the spearhead off, and the Mongol turned again, about to take an overhead swing, but the Spartan unsheathed his xiphos, and stabbed in the Mongol's shoulder. He fell down, and grabbed his glaive.
He sliced a few times at the Spartan, but each time it was deflected by the glaive. He eventually got into an unprotected area, and stabbed. The blade dodged the shield, and the tip penetrated his cuirass. But there wasn't enough power, and only got a small laceration on him. The Spartan threw down the glaive.
The Mongol reached at the ground and pulled something off his horse. His mace. He took heavy swings at the shield, and after a minute, there was a large crater in it. The Spartan threw the shield back up, knocking the Mongol down. The Spartan stood over him, and pointed his xiphos into the Mongol's stomach. He lifted it upwards, lacerating organs and muscles. He took it out, and stabbed the Mongol in the neck. "SPARTA!"
Bow & Arrow: 39.
Flanged Mace: 177.