Yes, this is puzzling. The subject can certainly come after the verb in Italian (see, for example, http://www.scuolitalia.com/1/gram-syntax-subjectafterverb.htm), but so can the direct object. However, unless the window is one of those animated objects in a Disney movie, it likely can't open itself, but is more likely to be the object being opened by an admittedly non-explicit and therefore ambiguous subject (io? tu? lui? lei?). I therefore think that 'la finestra' is almost certainly the direct object of apra.
Thank you for detailed analysis and it make sense. 'I don't want that window open' was the incorrect English translation I had. For this to be correct answer, I later thought, then Italian sentence should have been 'non voglio che si apra la finestra'. But now I wonder if using 'apra' in reflexive construction is correct?