A Brief History of the “House of the Rising Sun“ (2011)

Not quite. Even if the prison mentioned had existed for nearly half a century before the lyrics were composed, the War Department planned for early capture of New Orleans, which occurred about 15 months after Louisiana's succession and about a year into the war - relatively early. "Because a large part of the population had Union sympathies... the U.S. government took the unusual step of designating the areas of Louisiana then under U.S. control as a state within the Union, with its own elected representatives to the U.S. Congress."[1]

More relevantly, there never were any Confederate prisons for women. The first prison for women[2] wasn't established until 1873, at least 8 years after the Spring of 1865 when Lee courageously surrendered the last major Confederate army.

[1] https://en.wikipedia.org/wiki/Louisiana_in_the_American_Civi...

[2] https://en.wikipedia.org/wiki/Indiana_Women%27s_Prison