Not specifically on Arcturus, but I do have relevant information. Our galaxy has two major "populations" of stars, called "population I" and "population II". (I guess the Calvin and Hobbes cartoon was correct, scientists often use very unimaginative names). These correspond to "disk" and "halo" stars. Halo stars are distributed spherically around the galaxy center, and their orbits are fairly randomly distributed. Disk stars orbit in the plane of the galactic disk. If you pick at random some location in the galactic disk, then all disk stars near that point will have roughly the same velocity.

But there are a few nearby halo stars. These have large velocities in almost any direction (relative to our sun), whereas nearby disk stars have small velocities.
So perhaps you could try to find out if Arcturus is a population II star. There are other differences between Population I and II stars. A great general astronomy reference is Frank Shu's "The Physical Universe".

Well, I looked around but I couldn't find anything that seemed overly unusual about the magnitude of Arcturus' motion.Arcturus is a pretty close by (about 36 light years) red giant making it pretty bright. It also has a pretty large motion on the sky, but Barnard's star moves more and Sirius seemed to move in the same general direction. On the other hand, after talking to another answer, it turns out that Arcturus is a star that is from the galactic halo and passing through the disk of the galaxy near us. So it is not moving in the same general way as other stars that like the Sun are in the
galactic disk. Arcturus is also almost as close to us now as it's ever going to be.