Yes you can overcome a challenge, or you could use:
conquer, defeat, master, prevail against (or over) ,surmount, triumph over,
In the sentence that you give, the word challenge is used as a noun.
...
He conceived the project while he was on vacation.
An alternative word for "challenge" is "obstacle." It conveys a similar meaning of something that requires effort to overcome or deal with. Other synonyms include "difficulty" and "test," depending on the context in which you're using the word.
he fight them
Hannah does not overcome any challenges, rather she fails to overcome them. Like the challenge of deciding to live or die and the challenge to overcome the nasty rumors she's faced with.
Challenge accepted.
Example sentences: It was a challenge that she worked hard to overcome. (noun) The defense will challenge the testimony of this witness. (verb)
The biggest challenge for him is not sounding horrible and repetitive. It is a challenge he still hasn't overcome.
Fear is a challenge because the person is not going to want to do it since they are scared. If you can overcome the fear then you have won that challenge.
The final challenge that Zeus has to overcome before establishing his rule is defeating the Titans in a great war known as the Titanomachy.
Overcome means overwhelm: overcome, as with emotions or perceptual stimuli ; get the better of: win a victory over; "You must overcome all difficulties"; "defeat your enemies"; "He overcame his shyness"; "He overcame his infirmity"; "Her anger got the better of her and she blew up"