In the 1930's, I do believe the Alamo was a downtown business, maybe a drugstore?
It certainly wasn't out in a pasture somewhere.
It's a few blocks from the Riverwalk in San Antonio. Did cattle graze there in the 1910-1920 era? I have no idea. I do know from interviewing people for history projects that cattle grazed on the grounds of the State Capital in Austin in the twenties.
The song was written by Dmitri Tiomkin for the movie The Alamo in 1960. Tiomkin was a Russian Jew who emigrated to the US and scored movies like It's a Wonderful Life and High Noon. It's Hollywood, so there are doubtless inaccuracies.