Will the state where he has a warrant pick him up for a DUI charges if his case in another state is still open?

There is no Yes/No answer. It depends on the law of the state that has the open warrant. Usually, that charge is a misdemeanor or less and the state issuing the warrant is responsible for the cost of transportation; most states will not pay the cost of transportation for such a minor charge. Be aware that if he is arrested for any charge and the arresting state runs a computer check and sees an open warrant, the arresting state can hold him for a period of time to give the state issuing the warrant an opportunity to come and get him. Usually, the maximum hold is 30 days.