At the foot of the steps, the well-house is to the left, behind a metal grate:
NS3974 : Dumbarton Castle: entrance to the well-house. The well, in a very much older form, may have been the one that failed during the siege of 870 (see the article linked from the end-note).
According to the 1958 edition of the official guidebook to the Castle, the well was enclosed in the 1730s.
According to nineteenth-century local historian Donald MacLeod, in his "History of the Town and Castle of Dumbarton" (2nd edn, 1877), there was another source of water on the Rock, an "unfailing spring" on the eastern summit (the Beak), near the Magazine; I do not know what underlying geology would give rise to such a spring, but its location is marked on older editions of large-scale OS mapping. For a view of that area, see
NS4074 : Dumbarton Castle: the former site of a spring.
MacLeod does not mention the well-house as such, but I believe that he does discuss it in another role: when describing the structures on this level area, he says that "you may, however, take a peep at the blackhole, where military delinquents are confined".
My reason for making the identification of that "blackhole" with the well-house is that the slightly earlier writer John Glen says, in his 1847 "History of the Town and Castle of Dumbarton", that "immediately in front of the barracks, and under the guns and pavement, is the famed Dungeon, prison, or black-hole, where, in the earlier history of the Castle, state and other prisoners were confined at and previous to the Reformation". The barracks (see the first-edition OS map) were just to the north of the well-house. Glen and MacLeod do not mention the well-house separately, and there is a distinct possibility that they were referring to the covered well-house as the "black-hole" (another possibility, a less likely one, is that they were referring to some other structure that was cleared away when the barracks and other buildings between the peaks, except for the French Prison, were removed). See also
NS3974 : Dumbarton Castle: small chamber beyond the Portcullis Arch.
Immediately after that, Glen goes on to says that "the modern entrance is now from the south side", but the context reveals that he is there talking about the Castle as a whole, and not the "blackhole" (or even the adjacent barracks).
Just out of shot to the right is the French Prison:
NS3974 : Dumbarton Rock: the French Prison. These railings can be seen in front of the French Prison in another picture:
NS3974 : Dumbarton Rock and Castle.
In the nineteenth century, there were some other buildings nearby, to the south-east of the French Prison, namely, the Master Gunner's House and the Armoury. By this time, the Master Gunner's role was ceremonial, but not unimportant (see
NS3975 : The Drysdale Memorial); he would, for example, show visitors around the Armoury (which adjoined his house), and he would be rewarded with tips. For further comments on this, and on the buildings that used to stand in that area, see
NS3974 : Dumbarton Castle: level space between the peaks.
That level space is shown from the other side in
NS3974 : Dumbarton Rock: the French Prison.