I was almost overslept for the appointment of hair salon. I ran and arrived at the hair salon on time.

I eventually got my hair cut and dyed roots of hair. It looks like unique but I like it.

Does it sound natural?

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@haruruchun

Very well written.

I would write:
I almost overslept my hair salon appointment, but I ran and made it on time.

I eventually got my hair cut and my roots dyed. It looks unique, but I like it.

Explanations of everything I changed:
1) "oversleep" is a verb, so it's grammatically incorrect to say "I was overslept."

2) "for the appointment of hair salon" has article issues. As a style rule of thumb, we prefer to avoid longer "[noun] of [noun]" constructions in favor of using nouns like adjectives before other nouns. Another style rule of thumb is that we like to throw in possessives wherever we can, and this case is no exception. Another possible solution would be to say "for my appointment at the hair salon."

3) Since the two sentences are relatively short and since there is a good logical relationship between them (contrast), I decided to merge the sentences with a conjunction.

4) Stylistically speaking, learners overuse the term "arrived." A more common alternative in informal situations such as this one would be "made it." It isn't a mistake to leave it as is, just a point many learners aren't aware of.

5) Since you use the form "got my hair cut," it's preferable to have a matching structure for the roots part.

6) We say "it looks [adjective]." The word "like" is unnecessary.

7) I added a comma to split the last sentence up.