It sounds mostly correct to me, but I don't think it is expressed in the way a native speaker would express it.
There are two small errors, and I'll cover one at the end and one here. There is an error in your bold sentence in that to care isn't correctly conjugated. It should be cares because the subject ("the one") is singular third-person.
If I was to rephrase solely that sentence, I would say:
Because someone willing to do an arduous task for a person is
someone who really cares about them.
I think that "someone" is the better choice because the sentence is speaking generally, and isn't referring to any specific person. While "one" can be used in this way, I don't think that doing so is natural in this sentence. As a result, because the first part uses "someone," the second part can't refer to a specific person. It wouldn't be wrong to use "a person" in the second part, but because that is already used in the sentence it sounds better to use "someone" again.
As for the omission in my version, while "someone who is willing to" isn't incorrect, it's just more common and more natural to omit the "who is" part and say "someone willing to" (from Google Ngram Viewer):
Here is a comparison of dictionary definitions of "one" and "someone" (from Oxford Dictionaries):
A person of a specified kind:
"you're the one who ruined her life"
"my friends and loved ones"
"He's not one for making a fuss"
An unknown or unspecified person; some person:
"There's someone at the door"
"Someone from the audience shouted out"
"Some watches and cigars were also missing after someone forced a back
door to get in."
While you are referring to a person of a specified kind, the dictionary-given examples do not show similar usage. Further, in my experience, any uses of "one" in a manner similar to your sentence tend not to use the present tense: "The one who did this will receive swift justice." And, even then, I would expect "the person who..." to be used over "the one who...".
Whereas, when using the present tense, I have more often heard "someone" or "anyone" used than "the one": "Anyone breaking the rules will be punished" or "Someone choosing to break the rules will be punished."
I think that a better overall flow can be achieved by rewording more than just the sentence you've made bold. To better link "because..." to "it is worth it," you may want to use a conjunction instead:
Getting to the root of the problem and making it worth knowing is an arduous task. It takes time but it's worth it, and it's worth it because someone willing to do an arduous task for a person is someone who really cares about them.
Getting is a present participle, so making has to be too.
Using a conjunction here makes the sentence flow better because it more closely links the statement that getting to the root of the problem is worth it, and the reason for doing so being worth it.