Imo, speed doesn't need to be equalized. Eggman can create tech up to
Infinite in speed, and pilot it (Look at Klepto Mobile which is linked on his profile). He'd simply create stuff on Mega Man's level and pilot it. I assume his reactions scale to whatever he is piloting, otherwise I'm not very sure of how he'd keep up with Super Sonic in that fight. Obviously he won't be making Infinite speed tech here, but he will be creating tech (Vehicles, and Badniks) on Mega Man's level within a day.
As a frame of reference, Eggman has been able to mass produce robots on a island wide scale in less than a day (Sonic 1, 2, 3&K especially, & Chaotix), and has even taken over
a planet in less than a day (Little Planet, which was also completely mechanized and chained to a mountain on top of that),
and created Metal Sonic in that same timeframe. In 3&K, Eggman created Mecha Sonic Mk. 2 in a few hours tops (Mecha was created during the "& Knuckles" half of the game after his attempt to relaunch the Death Egg was foiled. Every level occurred during the day time) on top of still having an army capable of opposing Sonic in each Zone (Including Eggrobo boss-fights). I think it's also possible Eggman could hack a large amount of robotic forces from Dr. Wily or something to recruit to his army if need be.
So my guess is Eggman amasses a large army consisting of Badniks, Ships, Metal Sonic's (Possibly), and imo "Metal" Mega Man('s). My guess is he will also set traps to capture and disarm Mega Man with a day to research him. Also, with the mention of hacking, might it not be worth mentioning Eggman could unironically hack Mega Man? He
is a robot. While it is true he has a resistance, Eggman
does have feats of outhacking other supergenius' hacking (Tails) as well as defenses that are hard to breach by said other supergenius + Eggman's intelligence is greater than Dr. Light and Wily's (Bare in mind, Modern Eggman said Classic is = to him in intelligence. Just less experience).
I don't think he'd open up with hacking. But I think he'd try to as a
last resort if things were going bad. Also worth noting Eggman has fusionism via the Egg Mobile which can merge with and control technology it merges with, so that's a possible way to control Mega Man if he's losing badly.