Here is the full list of all 45 F1 2013 achievements. It takes around 30-35 hours to unlock all of the achievements on Windows.
Set clean lap times at 5 different circuits in Time Trial.
Set a valid lap time in One-Shot Qualifying.
Earn a medal in Scenario Mode.
Perform a Mid-Session Save.
Successfully complete all of the Young Driver Tests.
Complete all Day 1 tests in the Young Drivers Test.
Earn a gold medal in any of the Young Driver Tests.
Earn a gold medal in each Young Driver Test.
Take pole position in One-Shot Qualifying.
Complete a 100% race in Career or Grand Prix.
Complete a Career race without colliding with another car or object.
Finish a race in the top three in Career or Season Challenge.
Win a race with all driver assists disabled.
Complete 10 races using only the cockpit camera.
Take pole position in Career.
Complete 3 races in Split Screen.
Pass all of your objectives over a race weekend in Season Challenge.
Beat your first rival in Season Challenge.
Beat four rivals in one season in Season Challenge.
Stay with the same team through the whole season in Season Challenge.
Return to your starting team in Season Challenge.
Race for five different teams in Season Challenge in a single season.
Win the Drivers' Championship in Season Challenge.
Win the Drivers' Championship in Season Challenge on hard difficulty.
End R&D early on the current seasons car in Career.
Set the fastest lap time in every session of a long race weekend in Career.
In Career, qualify last, but finish first.
Move to another team mid-season in Career.
Win the Drivers' Championship in Career on any difficulty.
Win the Constructors' Championship in Career.
Post a time in all of the Time Attack scenarios.
Earn a gold medal in all of the Time Attack scenarios.
Earn a gold medal in every scenario in Scenario Mode.
Win the Drivers' Championship in Co-op.
Win the Constructors' Championship in Co-op.
Complete 25 online races.
Complete 50 online races.
Set a clean lap time for all circuits in Time Trial.
Earn a medal in each scenario of Rookie Driver in Scenario Mode.
Earn a medal in each scenario of Team Mate Challenge in Scenario Mode.
Earn a medal in each scenario of Drivers' Championship in Scenario Mode.
Earn a medal in each scenario of Final Year in Scenario Mode.
Win an Online Race with any car from the Classic 80's at Jerez.
Set a clean lap at Brands Hatch and Jerez in Time Trial with the Ferrari 312 T2.
Achieve gold in the Time Attack Classic 80's challenge.