I found a few entries in Thom's Dublin Directories that may be of interest - some of these directories include trades listings for various towns around Ireland.
1927 - Patrick Cagney, Castle Street - under 'Chemists and Druggists'
There's no street numbers used for Sligo town in this listing.
Other businesses on Castle street in the same listing include a James Currie, baker (also listed under confectioner); Bernard Carroll, boot merchant; John Tyler & Sons, boot merchant; James Cunlisk, butcher; J. & J. Graham, cutler (also listed as ironmonger); C.J. Mulligan, cutler; W.A. & A.F. Woods, cutler (also listed as ironmonger & leather merchant); F. & A. Egan, drapers; John Feeny, draper; McDonagh & Co., drapers (also listed as tailors); John Foley, grocer/tea/wine/spirit merchant; William Howley, grocer/tea/wine/spirit merchant; H. Clancy, hairdresser; F. Nelson, ironmonger; John Meldrum, oil & colour merchant; C.J. Mulligan, oil & colour merchant; J. Meldrum & sons, painters & decorators; James Galloway& sons, plumbers; James A. Flynn, printer; V. McVamaney, saddler/harness maker; J.A. Flynn, stationers & booksellers; Mrs. Herity, stationers & booksellers; and Francis Nelson, watchmaker.
Next towns listing I have is are 1932 and 1938, which also show Patrick Cagney at Castle street, and the next 1943/44 shows no entry for your Patrick, and the only 'Chemist & Druggist' entry for Castle street is a J. G. Smyth.
You can view a listing of all the occupied buildings on Castle Street on the 1911 census, including Patrick & Harriet at site ref. 17 (the street did not have numbers at that time - the census uses enumerator references). The House & Building returns (Form B1) included as part of the census include some basic details on the buildings - class of house, number of windows in the front, number of families
Harriet died in 1932, reported age 42, place of death Castle Street, occ. 'Chemist's wife', Patrick was the informant.
Edit : located a Trades listing for Sligo in a 1940 Thoms - no sign of your Patrick as a Chemist - the only one with a Castle St. address is that J.G. Smyth.