1. Have you tried asking the church? This may sound dumb but they probably do know how old their building is if this is what you want.
2. You could try looking for William Henry Shackleford's death certificate to confirm that their was a cemetery named Brooks Cemetery and that it was in Houston. You could also ask the church if you could look through some of their old records, because I know churches that have records from cemeteries on their property.
3. Again, go ask the church, or look up public records dealing with the piece of land at 5203 Fulton. Anyway you want, it will take some research.
Hope I could help, and hope you find what you are looking for. (What are you looking for?)