My brother took his ASVAB last week. Can the recruiter tell him what jobs are actually available for him when he joins?

As long as he's talking to an Army recruiter, the only way for him to join is by picking a job. It's not possible for him to join first and then find a job.
Other branches you can join without a job

The recruiter can inform him what MOS's are available now for him based on his ASVAB.

I believe that the only way he can find out for sure what jobs are available is by going through the enlistment process and sits down to pick a job when he goes to MEPS. The recruiter can give him an idea of what he is qualified for, but the availability of jobs changes frequently. MEPS will have the most updated lists.