If there is no gravity of the earth affecting the bullet or vice versa or friction in space, it all depends on the intial veloctiy of the earth and the bullet, whichever is intially faster will win, if they both start at 0 m/s then again it would all depend on their start velocity.
The bullet has a much much smaller mass than the earth, so the gravitational pull from the sun to the earth is goign to be huge compared to the bullets.
I havn't taken physics in awhile, but if the Force of gravity is stronger on the earth, I think it might accelerate faster around the orbit than the bullet but again I'm not sure if they are directly related or not.
Because if you think about it, the earth has a much greater chance of running into debree that could cause friction and slowly slow the earth down, or a big f'ing metor coudl nail it and the bullet would win hah.
v = (2*pi*R) / T,
if you use this formula, T is the period, R is the distance from the object from the sun, so mass doesn't even come into play, in this case they would be equal.
Well that was alot of rambling, I conclude that if they both start out at the same intial velocity, and there is no friction, they will both be equal in speed around the sun if you put treat the earth as a small point rather tahn taking into account how much bigger it is compared to the bullet.
Becuase somthing this big ( ) compared to . of course the right side of the ) will hit the finish line before the .
and here is a kitty: