As there are window 'lights' above the front door it is before electric light and so about 1880, the key stone aboive the windows suggests the need to show the street they were asporing to middle class which screams victorian, the narrow passage between the properties suggest they would have had to have access for the water closets in the back yard and so still 1880 - 1890.
Your landlord will surely know, his own buildings insurance company would want this info too (unless of course you have to insure the building?). Could you not just ask your neighbours? - they must need the same info for their own cover.
I would also suggest you walk up and down the street as there is probably ay least one date stone in the middle of a clock somewhere, if you five me the street address i can check the census for you too,
Why, as a tenant, would you need to know when the property you are renting was built? You only need to insure your contents not the building. Insuring the building is the landlord's responsibility.