Shadow is found right above the entrance to City Escape. Just stand on the words CITY ESCAPE, jump to the platform on the left, and hit the boost pad to land right next to him.
The fight itself is actually more of a race. As you run along the track, you'll see a giant purple meteor flying up ahead. That is an energy core. The first character that collects 2 of them get to use their special attack, which I'll get to later.
First off, boost. Always. Make sure to keep collecting rings and hitting the smaller blue meteors to keep your boost gauge charged up. Try to avoid boost ramps, as I've found being in the air really just slows you down.
While boosting, one of two things will happen if you run into Shadow. If he is also boosting, you will just bounce off eachother. If he isn't boosting, he'll get knocked down and drop rings. I'm not 100% sure if this counts as damaging him towards defeat though, and it doesn't happen too often, so don't plan on relying on that to defeat him.
Eventually, someone will reach the energy core. Whoever touches it first collects it. If you've been boosting non-stop and haven't hit any ramps, there's a pretty good chance you'll get there first. For whoever collects it, if it's their first, another energy core will spawn and you'll repeat the process. If it's their second core, they will go into their special attack.
When you get your special attack, you will perform a constant boost without pressing anything. Use the left and right bumpers to sidestep and ram into all the meteors you see on the track. They will shatter and fly straight to Shadow and hit him to slow him down a bit. Do this enough, and you will catch up to him. Then just get right behind him. You will make him drop all his rings, then throw him in the air. And that's it! achievement unlocked! If you somehow don't slow him down enough to catch up to him within about 30 seconds, the race for the energy cores will resume again.
Of course, sometimes things don't always go perfect, especially you first time through. So here's some stuff to remember.
If Shadow gets two energy cores first, he'll do his special attack. First, he uses Chaos Spear, which shoots a bunch of spikes into the ground ahead of you. Just sidestep them. After that, he'll use Chaos Control, in which he will shoot a huge purple laser at you. As he announces the attack, get ready to dodge. He'll say "Chaos..." and RIGHT before he says "Control!" sidestep to either side. After this, the race for the energy cores resumes.
During the normal race, if you let Shadow get too far ahead, he will use the same Chaos Spear attack to put spikes ahead of you. Dodge them in the same way.
EDIT: Added video guide. Credit to oxwcast for making it.