This should be rather easy to get. First, you need the Honda Legends Car Pack which is a FREE add-on. This gets you three cars: the 1984 Honda Civic CRX Mugen, the 1994 Honda Prelude Si, and most importantly the 2014 Honda Civic Si.
You'll still need to buy (or rent) the car with your credits, the add-on just adds it for purchase in your game. When you buy it you just need to drive it for 100 and 300 miles before March 31, 2014 to unlock the two related challenges. As far as I know these miles can be driven in ANY mode, but please correct me if I'm wrong.
The 2014 Honda Civic SI can be upgraded to D, C, B, and A class through quick upgrade. You'll need to do some special tuning, but you can get your Civic to S and even higher. This opens up a lot of events in the Class Competition leagues in Single Player. I also found the 2014 Honda Civic SI available for the Modern Hot Hatch in the Sport Compact League.
In Multiplayer, you can tune your car to A- or S-class and even drive in the Tag events for the other concurrent challenge:
You can use the quick upgrade option to get to A-class for the Pass/Keep the It tag events, but not S-class for the Virus tag events. Here's a quick guide to tuning your Civic to S-class. You can do one of 3 things:
a. Manually tune it - First quick upgrade to A-class then go to conversion and put in an alternate drivetrain to bump yourself up to S-class.
b. Search for an S-class user-created tune. There's plenty good ones out there.
c. Help a brother out and use my tune! Search 'xt5' in the description field for my tunes. I think I've got tunes up for all classes. I've also got a paint job under the same description if you feel so inclined.
If you were wondering about the special prize associated with the challenges, 10 people will win a $50 Xbox Gift card for completing the 100 mile challenge, and for completing the 300 mile challenge you are entered to win the grand prize of a trip for two to a performance driving school in Ohio and a Honda Mad Catz Xbox One Racing Wheel, of which there will be four winners.
Just for the sake of completeness, I'll be posting this solution on the other related challenge, so if you've read this you could skip reading the other one.