Which system of Korean numbers shall I use when counting hours in Korean?
I hear Koreans sometimes count hours by sino Korean numbers & sometimes they do by Native Korean numbers. Ex: Sometimes they say 이십사시간 (24hrs) & sometimes they say 두시간 (2hrs). that's why I'm confused.

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The sino Korean numbers are used if you are counting how many hours there are in a day. The Native Korean numbers are used for saying one o'clock, two o'clock, etc. Hope that helps^^

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I understand your confusion. There are some rules for reading numbers, but also many exceptions.
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Mostly before "hour" we use the native Korean numbers as in 1시 = 한 시, but before " second" we use the sino Korean numbers as in 30분=삼십 분.
As for 24 hours, we use both 이십사 시간 or 스물네 시간. I guess 이십사시간 is more common. ^^