This can be done in any game mode where you can complete a "full" Qualifying session (which contains Q1, Q2 and Q3).
An easy way to do it however is through Grand Prix.
Pick whichever car/driver/track you like.
In the race settings, you can change a few to make things easier/quicker:
> Weather: Clear
> Race Length: 3 Laps
> Rules and Flags: Off
> AI Difficulty: 0 (Very Easy)
Make sure that Qualifying is set to "Full" so that you get your 3 Qualifying Sessions.
Once you begin and are in the garage, open up the control panel (where standings, settings etc. are located) and fast-forward time until the session is over. As you didn't set a qualifying time you'll have failed to make it past Q1 and will start the race at the back.
From there, complete the race and finish within the top 3 positions to unlock the Achievement.