As with the other solutions, the Caterham is a good choice as well as the Ultima. Regardless, there is always the chance someone is going to have a faster tune and be a better driver.
The real information needed here (for those of you who believe you have done this but still have not resolved the challenge) is that there are two kinds of drag races. One apparently counts towards the challenge and one does not.
The first kind (that does not count) is where you drive into the parking garage and press Y to join an online free roam session.Having done that, you can drive down to the end of the long straight track where there is a blue icon that looks like a sprint race. This is where the drag race can be started. I did it this way half a dozen times, winning at least twice and did not get the challenge or the award car.
The second kind is the one that works. This is called a "Showdown". A Showdown is started by pressing X from inside the parking garage. You will see a message saying a showdown is starting and to press X to join. If there is no offer to press X to start a showdown, it means one is already in progress and you need to wait until it ends. Once it ends, there will be another opportunity to start a showdown. (The first person to press X starts the showdown and all there people pressing X after this will join that person - Thanks to SatNiteEduardo for filling in this piece). Once the showdown starts, just win it and the challenge will be satisfied for you will receive the award.
I used the Caterham with an S1 tune for this one. FWIW, There was also a Caterham with an A tune on the line. He did start first and he did get pretty far along before I started, caught up, and passed him with hundreds of feet to spare. However, I have also lost using this same car to a BAC Mono and an Ultima.
Note that either type of drag race described above counts towards the "Life's a Drag" achievement, so you'll want to keep track of which car classes you are using across either type of race.
(See Wyyvern's solution for "Life's a Drag" as he indicates there is a way for the player to initiate a showdown once the current one is complete. I have never seen this... I've only been offered the option to join a showdown that is already starting).