It's tougher, but we can. We know that the Sun has an orbital speed of approximately 220 km/s relative to the center of the galaxy. And we can measure the relative speeds of stars and gas clouds inside the Milky Way to derive a rotation curve for the Milky Way. This is discussed some here: https://ned.ipac.caltech.edu/level5/March01/Battaner/node9.h...
A side note that "speed of an arm" is potentially ambiguous. The arms in spiral galaxies are thought to be at least partly a reflection of density waves moving through disks. Those density waves can in principle have (pattern) speeds which are different than the orbital speeds of the gas and stars themselves. Talking about the speed of an arm could in principle refer to either the pattern speed of a density wave or to the velocities of the mass components which are overlapping with that density wave.