I'm sure I have come across this before, but I have never really stopped to think about it. I think the answer is pretty clear for anyone with some experience, but I find the question itself quite interesting and thought it might be of some value to someone.
The "accidentals apply only to one octave" rule is a bit misleading. Really, it should be "accidentals apply to the line (or space) they appear on". This is further illustrated by the C# half way through the measure, which, being at the start of the 8va section, is actually the same note as the first, yet it includes the accidental because it appears on a different place on the staff.
This is done to reduce clutter and improve clarity, thereby increasing the speed at which it can be read, while reducing the likelihood of errors. It is easy to see that removing the # from the lower C and adding a # to the higher C would make this passage much less clear.
So yes, the note in question should be played sharp.