Do we know if we are overtaking or falling behind our arm? Here's our galaxy as best we know, courtesy of this Wikipedia article:



It seems that as best we know, each arm is a compressed or brighter area (rather like a wave).


Looking at that image, do we know/guess if the arm-patterns are rotating? Or do they just sit there statically, unchanging from that diagram? Which way does it rotate, cw or acw per that diagram? (If we know.)
If the pattern is rotating - how long does it take it to rotate 360 degrees? As far as we know?


And hence ..


Indeed, is the arm-pattern catching us and moving through us ... or is the arm pattern slower than us and we're moving through it?
The answer to the question depends on precisely where you are in the Milky Way. According to density wave theory, which appears to describe the formation and evolution of spiral arms quite well, the arms move relative to some static inertial frame at a global pattern speed $\Omega_{gp}$. At some defining radius $R_c$, known as the corotation radius, stars move at $\Omega_{gp}$. Inside $R_c$, stars move quicker and thus move ahead of the arms; outside $R_c$, stars move slower and thus "fall behind" the arms.

There is uncertainty surrounding the values of $R_c$ and $\Omega_{gp}$ in the Milky Way in all its spiral arms - especially since local stellar kinematics differ in each of the arms, and so too may $R_c$ and $\Omega_{gp}$ - but measurements place the Solar System slightly inside the corotation radius, meaning that we are moving ahead of the arms.

I have data from a review by Gerhard (2011). Here are some basic average parameters, for the Milky Way as a whole:


Bar corotation radius ($R_{CR}$): $3.5$-$4.5\text{ kpc}$
Spiral arm corotation radius: $8.1$-$8.4\text{ kpc}$
Bar pattern speed ($\Omega_b$): $50$-$60\text{ km}\text{ s}^{-1}\text{ kpc}^{-1}$
Spiral arm pattern speed ($\Omega_{sp}$): $17$-$28\text{ km}\text{ s}^{-1}\text{ kpc}^{-1}$


This also goes along with a local mean speed of stars in the solar neighborhood - the Sun included - of $\sim220\text{ km}\text{ s}^{-1}$.



Hence for the three questions: (1) The pattern is rotating clockwise as seen in that diagram; (2) It rotates about once every 300 million years; (3) and at our radius, we're moving a little faster than the pattern so we are overtaking it.