Per Sage Advice a monk's unarmed attack is a melee weapon attack even though it is not a weapon. Champion Fighter's increased critical range is for weapon attacks, not weapons. Does that allow the monk to benefit from the increased critical range?

Yes, absolutely.

"Weapon Attack" is a specific term that covers physical attacks as compared to "Spell Attack".  It isn't restricted to attacks with a weapon.

Attacks are broken down into four basic categories based on two qualities: Melee vs Ranged and Weapon vs Spell. If an ability is described as augmenting attacks, it may or may not limit which attacks it affects by one or both qualities. For example, the Champion's "Improved Critical" affects all weapon attacks, both melee and ranged(so it doesn't work with something like Shocking Grasp). The "Crossbow Expert" feat benefits all ranged attacks, whether melee or spell(which makes it useful for spellcasters as well as archers). The Paladin's "Divine Smite" ability only affects melee weapon attacks(which is what Unarmed Strikes count as). And the Warlock's "Hex" spell affects ALL attacks. The Sun Soul Monk's "Radiant Sun Bolt" ability might not be an actual spell, but it's classified as a ranged spell attack for purposes of abilities like these that only affect certain types of attacks. 

It's also worth remembering that a "melee weapon attack" and an "attack with a melee weapon" are not always the same thing.