Yes you can overcome a challenge, or you could use:
conquer, defeat, master, prevail against (or over) ,surmount, triumph over,
Challenge (noun): a task or situation that tests someone's abilities or skills. Challenge (verb): to invite someone to engage in a competition or to test their abilities.
In the sentence that you give, the word challenge is used as a noun.
...
he fight them
Challenge accepted.
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.
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.
Not enough rainfall
The will to get things done
Type your answer here... completing what you started
As a noun: When you overcome a challenge, you can learn something along the way.As a verb: I will challenge him to a match to see if his skill matches his boasts.