Arcturus will not be disappearing in 300 years.
However, your broader question is this: When a star is moving, do we see where it is, or where it was when it emitted the light that we are seeing now?
The answer is that we see light that the star gave off, and we see that light coming from where the star was.
Similarly, when I was a boy, I used to see low flying military jets flying over head on a regular basis, and I could hear their noise, but the noise was always seemed to be coming from about a mile behind the aircraft. This is because I heard the plane where it was when it made the noise, but by the time the noise got to me (about 5 to 10 seconds later) the plane had moved on.
Forming opinions as we speak