Firstly, if your brother is planning to stay for a week it's not a transit visa that he'll need, but an entry permit.
To get an entry permit he'll need a sponsor in the UAE. If you are a permanent resident then you can act in this capacity. Otherwise the UAE airlines and hotels can do so.
Once you're ready to apply, you do so through the GDRFA who have several offices throughout the region. You can find full detail of the visa requirements, airline visa services, application procedure and links to the GDRFA offices on the UAE government web site