Bendywendy is right. Chariot Street, or what's left of it, is still there linking Paragon Street and Carr Lane. But it's at the western end of the City Hall, not the Maritime Museum. It's only a few yards long now. Before WW2 it continued North West under what is now Queens House. The Hull Cheese used to called the Paragon Inn and was certainly there (or its predessor was) when Aaron was in the area
I've had a quick look at the Hull History Centre catalogue
http://lib3.adir.hull.ac.uk/dserve/search.htm and searched for Chariot Street. This turned up a few hits, mainly the deeds and similar things, nothing for No.19. There are some photographs dated 1941 and 1942 so I guess they are of the damage during the blitz.
I also searched on Aaron Baker. That turned up something interesting. In document ref C TCC/6/3, Admissions to Hull Charterhouse - Aarons name is listed followed by "(letter of thanks)". As you know Aaron and Hannah were listed in the 1871 census as living at the Charterhouse so I wonder if that is a letter written by Aaron at the time they moved there. Intriguing isn't it! Depending on the time you've got when you visit Hull it might be worth popping into the History Centre and having a look at that document and perhaps some of the others. If you contact them in advance (24 hours) they can have the documents ready for you.
Dave