According to its entry at the great BhamWiki site, this hotel was developed by D.M. Rush around 1919 and operated until about 1949. The 1945 Birmingham Yellow pages gives its address as 316 1/2 North 18th Street and the phone number was 7-09411. The small hotel occupied just the second floor of the building seen in the photo below, and was one of the few such accommodations available to blacks in the city.
In the 1930's the facility was owned by Tom Hayes. The place apparently provided hotel arrangements for visiting teams in town to play the Birmingham Black Barons, which played professional baseball in the Negro leagues from 1919 until 1960. In his book Black Baseball's Last Team Standing: The Birmingham Black Barons, 1919-1962 [2019], Bill Plott has a couple of tidbits about the hotel. Manager of the Barons in 1938 was William "Dizzy" Dismukes, a Birmingham native who had also managed the team in 1924. Anyone who wanted to try out for the team that year could contact him at the Rush Hotel [p.117]. He also writes that in 1953 Dismukes had set up shop again at the hotel, this time to recruit for the New York Yankees [p.221]. Thus the hotel may have operated past 1949.
A man named Joe Rush was Black Barons owner in 1923 and 1924, and owner and president in 1925. The Birmingham Black History Project on Facebook has more information about the Rush family.
Below I've also included a photo from Google Street View showing what the building looked like in 2019.
Photo by W.B. Phillips in 1931
Source: Birmingham Public Library Digital Collections
Here's the building via Google Street View February 2019. You can see the same arch over the doorway and the eight windows in the second story.