No. Living person can not hire someone to perform obligatory prayers or obligatory fast on his behalf. No one can perform such obligatory prayers and fast on behalf of any living person.
It is only Hajj if the living person is absolutely disabled and completely unable to perform Hajj in person even by being carried by others, then it will be permissible for him to hire someone to perform obligatory Hajj on his behalf.
Any one can hire some one to perform recommended Hajj or Umra or Ziyarat on his behalf. Ali Ibn Yaqteen who was a pious companion of Imam Musa Al-Kadhim (AS), used to hire 25 persons every year to perform recommended Hajj on his behalf.