If the sister of my husband will file I-130 for him, and my husband is in the US under H4 visa can he adjust his status so he can stay in the US?

Your husband cannot stay in the U.S. and adjust status just because an I-130 petition for him was filed. Every year, a limited number of immigrant visas is made available for relatives of U.S. citizens and permanent residents.The number of people who file I-130 petitions is far greater than the number of available visas. As a result, you have to wait until all the people for whom I-130 petitions were filed before yours would get their immigrant visas. In immigration practice, it is called waiting until your priority date (the date of filing of I-130 petition) becomes current. The waiting time is different for different categories and for citizens of different parts of the world. Your husband, as a brother of U.S. citizen will have to wait for his immigrant visa about 12 years (17 years - if he is from Mexico, 24 years - from Philippines). Only after his priority date becomes current, your husband will be able to apply for adjustment of status (if he will be - legally! - in the U.S.) or receive an immigrant visa through the U.S. consul in your country. Staying in the U.S. after expiration of your visas will make it difficult (or even impossible) to adjust status or to receive immigrant visas when your husband's priority date becomes current.

The sister's petition will take at least 10 years for the priority date to become current.