I think napping is really good for one's health...

Here, "napping" refers to the act of taking a nap in general.

No sooner had I woken up, than I thought napping is so good...

(1) The complete grammar construction is "no sooner had A than B".
(2) Again, "napping" refers to the act of taking a nap in general.

My brain is so clear, and what is more, a little of my fatigue flyedhas flown away.

(1) I'm happy to see that you've learnt to use "a little" for uncountable things. Previously, it was used before an adjective ("short"). However, when used before a noun ("my fatigue"), it becomes "a little of".
(2) "Fly" is an irregular verb. Its past tense is "flew" (used to descibe a past event) and its present perfect tense is "has flown" (used to describe a present state).
(3) Alternatively, instead of "has flown away", you can also say "has disappeared" or "has vanished". These may be closer in translation to 取れている, but they are less figurative.

Although I have an exam that is really important forto me tomorrow, I was badfelt ill.

(1) Alternatively, you can simply write "Although I have a really important exam tomorrow".
(2) The phrase is "important to (someone)". The phrase "important for" exists, but it is used in different contexts. If you are curious, I can provide a brief explanation of "important for" as well.
(3) "I was bad" is an extremely vague phrase. I think "I felt ill" is a good translation of "体調が悪かった（んです）".

It may be caused by nervousness.

"Nervous" is an adjective. The noun is "nervousness" or "nerves". (These two words are synonyms).

Although I understood I wasand (that) I ought to have studyied for the exam, I did what iwas necessary for me!

The tenses at play here may be a little confusing. I will explain each correction individually.

(1) For "I understand", the present tense is used. This is because you're referring to your PRESENT state of understanding. In the native text, you also wrote "分かっています". "I understood" would translate to "分かっていました" instead.
(2) For "ought to have", the past tense is used. This is because you're saying you should have studied in the PAST. In the native text, you also wrote "すべきだった". "Ought to" would translate to "すべきだ" instead.
(3) For "what was necessary", the past tense is used. This is because taking a nap was necessary in the PAST. This is similar to (2).

I think health is the most important!

I think omitting the "the" will make the sentence more natural. This is because when we see "the", we expect a noun. For example: "I think health is the most important thing in life." Here, "thing in life" is the noun.

No sooner had I woken up, I thought the nap is so good...

My brain is so clear, and what is more, a little my fatigue flyedew away.

Although I have an exam really important for meexam tomorrow, I was bad.

It may be caused by nervousness.

Although I understood I was ought to study for the exam, I did what is necessary for me!

I think a nap is really good for one's health...

You could also say, " I think napping...."

No sooner had I woken up, I thought my nap iwas so good...

You could also say, " I thought napping is so good..." or "naps are so good..." Saying " is so good" is also a bit awkward, but not incorrect. I would say instead, "I thought my nap (the nap or napping) was great.

My brain is so clear, and what is more, a little my fatigue flyedhas flown away.

Although I have an exam really important for meexam tomorrow, I was bad.

In English, we put all of our descriptor words before the noun.

It may be caused by nervous.ness

Although I understood I wasthat I ought to study for the exam, I did what is necessary for me!

I think health is the most important!