Is the rotation of Gravity Probe B's frame dragging precession in the same direction or opposite direction of Earth's rotation? First, thank you to the community for answering questions like this, it is much appreciated.  Second, I did try to find the answer online first, and I believe from some images that the direction of rotation about a North-South axis line is the same as that of Earth, but I could not find any definitive statement to that effect.
That is, looking down on the Earth from above the North Pole, our planet rotates counter clockwise.
I believe that, viewing from this position, the GP-B satellite rotates counter clockwise (same as the Earth) when above the North pole, and clockwise (the opposite of the Earth) when above the Equator.  The net effect is a rotation of about 39 milliarcseconds per year as described at https://arxiv.org/ftp/arxiv/papers/0802/0802.3346.pdf.  I am just double checking - Is that net rotation counter clockwise (same as Earth) as viewed from above the North Pole?
 A: Yes, you are correct at the North pole and at the equator. As shown by eqn 1.10 of the arxiv paper you reference:
$$
\vec{\Omega} = \frac{G}{c^2}(\frac{3\vec{S}*\vec{R}}{R^5}\vec{R}-\frac{\vec{S}}{R^3})
$$
where $\vec{\Omega}$ is the angular velocity of precession of the test mass, $\vec{S}$ is the angular momentum vector of the earth and $\vec{R}$ is the vector from the earth's center to the satellite. At the north pole $\vec{S}$ and $\vec{R}$ are in the same direction so $\vec{\Omega}_{NP} = \frac{G}{c^2}(\frac{2\vec{S}}{R^3})$ while at the equator $\vec{S}$ and $\vec{R}$ are perpendicular so $\vec{\Omega}_{EQ} = -\frac{G}{c^2}(\frac{\vec{S}}{R^3})$.
As you say, the GP-B satellite rotates the same as the Earth when above the North pole (and the South pole), and the opposite of the Earth when above the Equator. The satellite is in a polar orbit.   The integrated precession over the orbit is -39 mas/yr which is in the opposite direction to the earth's angular momentum, and therefore is clockwise when looking down on the North pole.
