I passed this Mission pretty quickly the next day, as I thought I would. I knew what time gap I needed by the first checkpoint, and so could quit at that point if I'd messed it up. Essentially, the fifth-place car needs to be clearly in sight at the first checkpoint, and then you begin threading your way past them. Staying in fifth gear (resisting temptation to shift up to sixth) seemed like it might have helped on the winning run.
I think this Mission should be easy for most people; master of Fuji or no.
Essentially, you have three corners to get right, all by yourself with no traffic in the way, and then you get close enough that you can just power pass the other cars.