Howdy skysurfer,
You are likely correct. The biggest issue in space is being able to lose excess heat by radiation. At the point you mention, the spacecraft would be within the corona, and enduring very high temperatures (the Sun's corona is around 10^6 degrees C). Heating would be via radiation and conduction. The influx of heat would need to be re-radiated into space or the craft will boil inside.
Arcturus is a red giant, and so has *slightly* cooler surface temperatures than our sun (4300 K vs 5800 K). This difference is not appreciable to humans - it's entirely too warm to survive for even a second.
The radiation flux is even more lethal.
Isaac Asimov wrote a very good SF story which passed on this topic - Lucky Starr and the Pirates of the Asteroids. It provided a good rundown on the science, in a very entertaining way. Asimov made use of a perfect mirror for the outside of the ship to reduce the influx of radiant heat.
In real life, NASA is sending the space probe "Parker" to the sun sometime in 2018. It will be much further away from the sun (10 solar radii) than Elite Dangerous would appear to be from Arcturus.
http://www.abc.net.au/news/science/2...he-sun/8572540
http://parkersolarprobe.jhuapl.edu/S...e-Environments