According to the Texas Code of Criminal Procedure, here's the definition of an arrest warrant:
Art. 15.01. WARRANT OF ARREST.
A "warrant of arrest" is a written order from a magistrate, directed to a peace officer or some other person specially named, commanding him to take the body of the person accused of an offense, to be dealt with according to law.
So the fact that the city he was stopped in has no jail would not get this matter dismissed. He could still be picked up anywhere in the state, held on the warrant and transported to court to face the magistrate who issued the warrant. Then, if he could be sent to county court for contempt.
Many people have warrants for failing to pay traffic fines. And the only way out of the situation is pay the fine.
Is this a traffic matter? And if so, are these unpaid tickets, or an offense that was filed with the court on which he failed to appear? If the former, he may be able to go to the DMV and set up a payment plan for the tickets. In my experience, the DMV will frequently even waive some of its penalties and late fess if they know they will have a better chance of getting the rest of it. If the latter, he'll be expected to come before the judge who issued the warrant and deal with his case.