During my RTS battles with the Drowning Doom, those god damn glowing blue reaper horsemen are giving me serious woes.  Does anybody know what the most effective anti reaper unit is?  They seem to cleave through whatever I throw at them, including Eddie.

I think Bouncers do a decent job against em

Spawn the fire biker dudes to keep them at bay. I had the same problem, and make sure to keep pressing forward in this one battle in particular.

It's the bridge fight.  My plan is to throw all I can spawn at them for as long as I can.

Upgrade the razor girls and double team, they kill everything, Fire Barons could also take them out easily.

You're best going long range against reapers, so maintain and upgrade a small force of Razor Girls or get a Skull Splitter on the field and double team whenever you see one appear.  Eddie's guitar could do the job eventually if the reaper's not focused on him.

You can upgrade units?!  How?!

LOLS!  I just rushed her stage with about four or five groups of Razors, and about 1000 Roadies.  It must have been amazing for the Drowning Doom to see this giant flock of migrating speakers I was protecting come hell or high water.

"Upgrade the razor girls and double team, they kill everything, Fire Barons could also take them out easily. "

I was going to say use the Bolt Thrower upgrade and ride your car into Battle. It easily takes out most Doom units that aren't vehicles, and keeps pesky commanders from killing all your dudes, through fierce knockback. 
 
On the other hand, if you wish to purely use units, then yeah, upgrade and double team with your Razor Girls and Fire Barons.