Warning -- this is probably waay more than you ever wanted to know about this building's past. I finally got a spare hour at the main library Monday, and poked around the Polk City Directories. (I love them -- and have spent a lot of time finding out the history of my variouss apartments) I didn't have as much times as I would have liked, and didn't go back all the way or check every year. But this is what I found:
The first listing I got for 1533 Mt Vernon was in 1923-1925, for the "Accurate Measure Oil Co (sta)". But when I looked that name up in the business portion of the directory to try and get more detail, it gave the company's address at 1884 East Broad, and that's where it was in 1926, when this photo was taken. So... who knows.

Between 1926 and 1932 #1533 was occupied by Ben Golden, barber, and #1537 was (Hubert) Kanmacher Drugs. By 1929 there was a Shell Petroleum gas station on the nw corner of that intersection with Taylor, that seems to have changed hands but stayed in business until at least the 80's.
In 1933, the pharmacy changed hands and became (Clifton E) Workman Drugs. And in 1938 it changed again, to (Howard E) Lemmon Drugs, then Lemmon's Pharmacy. Ben Golden, barber, persevered in 1533.
There was a big gap in the directories on the shelf Monday; the next volume I found was 1947. Lemmon's Drugstore was still there, but Ben Golden had expanded to "barber and billiards", which is how it was listed in 1951 as well. (Talk about a nice commute; he and wife jeanette lived around the corner at 332 Taylor.)
Another big gap in the books, and the next listing, for 1958, had a whole new cast, altho' similar businesses. 1533 was occupied by Geo L Jameson, barber, and 1537 was occupied by James Pharmacy. But listed without addresses in between the two were Robinson Insurance Agency, and J&R Cin. (No idea was 'Cin' means.) The following year, the building was occupied by Jameson's Barber Shop, Jameson's Cins, and James Pharmacy. And by 1960, only Jameson Cins and James Pharmacy. In 1961 1533 was vacant, but James Pharmacy was still there. In the volumes for the 50's and 60's I first started noticing regular listings for four apartments in the rear of the building. and I was kinda surprised to that the East Market was still around.
In 1537 James Pharmacy hung on thru the 1960's. 1962's Speedy's Laundramat gave way to Speedy's Janitorial Service in 1964, but was back to a laundramat in 1965 and for the rest of the decade. 1966 was the first time I saw a listing for 1535 -- the home of machine operator John Hill, who lived there for another decade or so.
After another gap in the record, the 1974 book listed 1533 as Gentry's Laundry, there were 5 individual's named at 1535, and Lucy's Restaurant occcupied 1537. Same roster in 1975.
Another gap, but all adresses were listed as vacant from 1981-1985. In 1986 and 87-88 1537 was occupied by Pheonis West Art Service "ret sls painting" -- an art gallery?
Seems like there hasn't been anything in there since. Of all the businesses there the one I like imagining best is "barber and billiards" Betcha get a cool view out of those 2-3 story windows.