The Goad map shows a couple of warehouses near no.2 High Street, and also handy skylights which may have been the means of entry. There seems to be an empty lot at the back, however by 1894 there are a couple of buildings there between Duke St and Hanover Place. If these had just been built in 1888, could they be the empty buildings where the thieves got in?
Another snippet from The Star 1st Oct reveals that at least some of the vacant lots in Duke Street shown on the 1887 map had been built on by 1888;
"Mrs. Lindsay, who occupies the two front rooms of 11, Duke-street - almost opposite Church-passage, leading to the court"
Number 11 didn't exist in 1887, the map shows a blank space between 8 and 14. Likewise nothing is shown between Aldgate and no.7 but if these had all been redeveloped (as per the OS map) by the time of the Eddowes murder then there would have been a building on Duke Street behind the new Post Office.
Aha! This report from the Daily News 2nd Oct seems to confirm my theory that the empty building may have been newly built;
Nevertheless, the police seem certainly to have been caught napping in a manner which yesterday morning appeared to afford very general amusement to the crowds assembling in and about Duke-street. While the spot was literally garrisoned with police, and everybody's attention was absorbed in the one great calamity, it was found that burglars had taken advantage of the occasion to slip into unoccupied new premises in Duke-street and work their way thence into the post-office fronting Aldgate and commit robbery to a considerable extent. Whether this was effected before the excitement over the murders arose, or as quite conceivable in the very midst of the general agitation and absorption of attention, we have at present no information.
Also, Philip Sugden's "Complete History" mentions in the index that the Post Office was between Duke St and Houndsditch, although sadly he doesn't seem to say how he found this out.
"The Aldgate post-office backs upon the rear of some newly-erected and untenanted warehouses in Duke-street. On Saturday evening at half-past eight o'clock everything was safe, and the premises were locked up. Yesterday morning at eight o'clock, when the staff resumed duty, it was discovered that either on Saturday night, or Sunday night, burglars had affected an entrance and broke open the tills, from which they took stamps of all values, from ½d to 10s, to the total amount of £260, besides £50 in cash. The cellar lights of the warehouses in Duke-street were broken, and it was in this way that the thieves obtained access. Then they scaled the roofs, and got through a trap in the upper part of the post-office. They crept downstairs, and in order to avoid being seen through the windows opening into the street they judged it necessary to work their way to the apartment where the safe was placed in a roundabout manner. They pulled up a part of the staircase, and went through the cellar. Apparently they were engaged several hours upon the place, as they had to force open some drawers. Before going away with their booty they carefully washed their hands. In its connection with the investigation of the murder near at hand, the fact that the post-office was entered by way of the insecure premises in Duke-street shows that the burglary must have been committed on Saturday night, for throughout Sunday the street was thronged with people."