There are two ways to read numbers in Korean. One is to read Chinese character in Korean way and the other is to read in pure Korean way. When used in adjective forms, the latter changes while the former doesn't. Let me distinguish them for convenience.
(A) Chinese: 일, 이, 삼, 사, 오, ...
(B) Korean: 하나, 둘, 셋, 넷, 다섯, ...
(C) adjective Korean: 한, 두, 세, 네, 다섯, ...
When speaking Korean, we need to know which type should be used. When writing Korean, we often replace numerals with Arabic numerals. I don't know any written rules stating which type can be replaced. However, I am sure that it's okay to replace numerals of type (A).
In fact, I insist that Arabic numerals should always be used for type (A), unless you are writing literary articles. Korean sentences are sometimes ambiguous because there are many short words of different meaning with the same pronunciation. The use of Arabic one reduces the ambiguity. For instance, 이년 can be interpreted as '2 years' or 'this bitch'.
The situation is subtle for type (C). About 20 years ago, my Korean teacher claimed that Arabic numerals must be read as type (A). Although school teachers are not very reliable and languages change over time, I partially agree with him. But if we forbid the replacement for type (C), there is a problem. Type (A) and type (C) are not disjoint.
As we see in the answer by Vladhagen, it's common to use Arabic numerals to address time. Let's change the time for explanation.
5시 5분에 일어났어요.
다섯 시 오 분에 일어났어요.
It's weird but most Koreans use type (C) for hours and type (A) for minutes. Maybe it's because numbers used for minutes are bigger. Compare this with the following.
one apple: 사과 한 개(correct and common), 사과 일 개(very weird, maybe correct but not sure)
fifty apples: 사과 쉰 개(correct but not common), 사과 오십 개(correct and common)
In my opinion, there is no problem unless you use both Korean and Arabic in the same sentence. (다섯 시 5분 is a little strange.)
On the other hand, never use Arabic numerals for type (B). Look at the following sentences.
하나도 이해하지 못했다. (I couldn't understand a single thing.)
1도 이해하지 못했다.
I am not as stubborn as my teacher, but the second sentence is quite annoying. If I see such sentence, I get an impression that the writer is a kid enjoying using broken Korean.
In conclusion, my suggestion is the following.
(A) Use Arabic numerals.
(B) Never use them.
(C) Avoid using them if the Korean numeral is short hence not cumbersome.