This page of the Mafia Definitive Edition guide contains a walkthrough for Mister Fancy Bike side mission available in the Free Ride mode. You will learn where to find the phone box next to Carlo's apartment and how to pass the time trial using a motorcycle.
You can learn about this mission by finding Herbert Baskerville's letter in the Free Ride. The letter is found on the large table in the room upstairs (the one for planning assassinations).
It has information on where the mission will begin. Finding the pay phone can be problematic. The game hints that it is next to Carlo's apartment, the one from the Bon Appetit! main mission . However, you have to go south of the block, to the southern edge of the Holbrook district, close to the Downtown.
This mission is a time trial and you are driving a motorcycle. You have to cross a route marked by checkpoints. Reaching each checkpoint adds 10 seconds to the timer. If you drive flawlessly, you will be able to reach the next checkpoints in less than 10 seconds, so there is a large margin of error in case of a crash or a wrong turn.
The key to winning this time trial is high precision. Do not try to drive as fast as possible, as the route involves passing through many narrow places and making many sharp turns.
Drive faster in those places where you have to jump and "touch" a checkpoint hanging in the air – the picture shows an example of such a jump spot. It's also worth going up the stairs at a higher speed so you don't stop at them.
For the rest of the route, you must drive on, e.g. the pavement that runs along the bridge. You will also have to make several sharp turns. Don't drive through checkpoints too fast so you don't have to turn around when you miss a sharp turn.
The reward for completing this mission is Waybar motorcycle. From now on, you can take it from any garage on the map. The game also allows you to change the color of the body. It is a great machine that can move faster than a regular motorcycle.