here you go, playing 4,5,4,5,4,5,fingers....for a while and perhaps for few years and treated as a dessert for every practice session-including other major works.
I disagree - I don't play chromatic octave runs with strict alternating fingers 4 and 5. Because it is not ergonomic to use the 5th finger on a black key and the 4th finger on a white key what would happen this way. Instead, as others wrote already, use the finger 4 for black keys and finger 5 for white keys. Or better, if you have large hands, use also finger 3. E.g., right hand starting from c to the next higher octave: c: 5, c#:4, then 5,3,4,5,4,5,4,5,3,4,5 (next c). This way a lovely legato is possible (unfortunately not for the other notes played with the thumb), and with time it can get very fast too without creating unnecessary tension in the wrist.