tl;dr Stream of consciousness, partly; interior monologue, no.
A Portrait of the Artist as a Young Man is narrated largely through indirect discourse limited to Stephen Dedalus's point of view. Inasmuch as the narrative technique is third person, the novel does not use interior monologue, which by definition uses first person. However, as the question correctly notes, the narrative point of view develops in tandem with Stephen as he grows from a very young child to a young man. In the first chapter, when the child Stephen has little cognitive self-awareness, Joyce manipulates free indirect discourse to present Stephen's experiences in a way that approximates a child's stream of consciousness.
The very first words, quoted in the question, are a story Stephen's father tells him. The chapter continues:
His father told him that story: his father looked at him through a glass: he had a hairy face.
He was baby tuckoo. The moscow came down the road where Betty Byrne lived: she sold lemon platt.
[Joyce, James. A Portrait of the Artist as a Young Man. 1915. Penguin Great Books of the Twentieth Century. New York: Penguin, 1999. p. 1].
The childish diction and the leaps of logic indicate that this is the very young Stephen's way of experiencing the world. Rather than interior monologue, Joyce uses free indirect discourse here. It's indirect in that Stephen's experiences are narrated in the third person rather than directly quoted. It's free in that there are no markers such as "Stephen's father said" or "Stephen thought". This technique does serve Joyce well in representing the young Stephen's stream of consciousness.
Here is a clearer example of Joyce's use of free indirect discourse to represent Stephen's stream of consciousness. Stephen has just heard a schoolmate call another a "suck":
Once he had washed his hands in the lavatory of the Wicklow hotel and his father pulled the stopper up by the chain after and the dirty water went down through the hole in the basin. And when it had all gone down slowly the hole in the basin had made a sound like that: suck. Only louder.
To remember that and the white look of the lavatory made him feel cold and then hot. There were two cocks that you turned and water came out: cold and hot. He felt cold and then a little hot: and he could see the names printed on the cocks. That was a very queer thing.
And the air in the corridor chilled him too. It was queer and wettish. But soon the gas would be lit and in burning it made a light noise like a little song. [p. 5]
Memory, sensory experience, emotional state, and cognition are bound up together: Stephen remembers washing his hands, and re-experiences the cold and the heat of the water from the two faucets. The sensations in turn trigger another memory, that of seeing the cold and hot labels. Simultaneously, Stephen finds it queer that he's actually remembering the event so vividly, to the point of remembering the labels. His thoughts then return to the present, because he realizes that the corridor is also "queer" and "wettish", like his memory. He then thinks of the immediate future, when the gas will be lit. The free association between words, images, and sensations; the lack of linearity in the handling of time; the evocation of sense perceptions such as sight, heat and cold, and hearing; and the intermingling of memory and cognition with all of the above, together present Stephen's stream of consciousness.
But as Stephen grows older, Joyce moves away from free indirect discourse to a more traditional third person limited point of view. We still see the world through Stephen's eyes, but the narration is much more straightforward:
The priest's face was in total shadow but the waning daylight from behind him touched the deeply grooved temples and the curves of the skull. Stephen followed also with his ears the accents and intervals of the priest's voice as he spoke gravely and cordially of indifferent themes, the vacation which had just ended, the colleges of the order abroad, the transference of masters. The grave and cordial voice went on easily with its tale, and in the pauses Stephen felt bound to set it on again with respectful questions. He knew that the tale was a prelude and his mind waited for the sequel. [p. 131]
Unlike in the first chapter, what Stephen sees and hears is presented quite objectively. Considering this passage in isolation, there would be little that differentiates it from an omniscient narrator point of view. Stephen's thoughts are indicated with words like "he knew" or "felt bound to". By this point, Joyce's technique no longer approximates stream of consciousness.
The closing section of the novel, a series of diary entries from Stephen, mark yet another change in the narrative technique. We have moved from free indirect stream of consciousness at the opening, through a traditional third person limited point of view for most of the novel, to a first person narrator at the end:
26 April: Mother is putting my new secondhand clothes in order. She prays now, she says, that I may learn in my own life and away from home and friends what the heart is and what it feels. Amen. So be it. Welcome, O life! I go to encounter for the millionth time the reality of experience and to forge in the smithy of my soul the uncreated conscience of my race.
27 April: Old father, old artificer, stand me now and ever in good stead. [p. 217]
In a neat bit of self-referentiality, Joyce discusses this modulation and development of the technique in the novel itself. Stephen and his friend Lynch are discussing the three modes of literature: lyric, epic, and dramatic. Stephen says:
The lyrical form is in fact the simplest verbal vesture of an instant of emotion, a rhythmical cry such as ages ago cheered on the man who pulled at the oar or dragged stones up a slope. He who utters it is more conscious of the instant of emotion than of himself as feeling emotion. The simplest epical form is seen emerging out of lyrical literature when the artist prolongs and broods upon himself as the centre of an epical event and this form progresses till the centre of emotional gravity is equidistant from the artist himself and from others. The narrative is no longer purely personal. The personality of the artist passes into the narration itself, flowing round and round the persons and the action like a vital sea. This progress you will see easily in that old English ballad Turpin Hero which begins in the first person and ends in the third person. The dramatic form is reached when the vitality which has flowed and eddied round each person fills every person with such vital force that he or she assumes a proper and intangible esthetic life. The personality of the artist, at first a cry or a cadence or a mood and then a fluid and lambent narrative, finally refines itself out of existence, impersonalizes itself, so to speak. The esthetic image in the dramatic form is life purified in and reprojected from the human imagination. The mystery of esthetic, like that of material creation, is accomplished. The artist, like the God of creation, remains within or behind or beyond or above his handiwork, invisible, refined out of existence, indifferent, paring his fingernails. [p. 184]
The first chapter, with its free indirect discourse approximating the child Stephen's stream of consciousness, exemplifies the lyrical mode, with Stephen "more conscious of the instant of emotion than of himself as feeling emotion". The bulk of the novel is in third person limited, "when the artist prolongs and broods upon himself as the centre of an epical event and ... [t]he narrative is no longer purely personal"; this is the epic mode. Finally, at the end, with our reading Stephen's diary entries, it is as though we no longer have an intervening author between us and the character: "The personality of the artist, at first a cry or a cadence or a mood and then a fluid and lambent narrative, finally refines itself out of existence, impersonalizes itself, so to speak". We have arrived at the dramatic mode.
The reference to Turpin Hero is also noteworthy here. The original version of A Portrait of the Artist as a Young Man was called Stephen Hero. And whereas the ballad, as Stephen remarks, begins in the first person and ends in the third, Portrait has the opposite trajectory. Joyce's later novels, Ulysses and Finnegans Wake, are acclaimed for his formal experimentation, particularly in stream of consciousness techniques. But Portrait, considered a minor work in comparison to the later monuments, nevertheless demonstrates Joyce's interest in and control over narrative form and technique.