The passage in context is the narrator's arrangement of two separate events. The first (1:29-31) is a response to seeing Jesus. The second (1:32-34) is the Baptist's witness of Jesus' mission:
29 The next day he saw Jesus coming toward him, and said, “Behold, the Lamb of God, who takes away the sin of the world! 30 This is he of whom I said, ‘After me comes a man who ranks before me, because he was before me.’ 31 I myself did not know him, but for this purpose I came baptizing with water, that he might be revealed to Israel.” 32 And John bore witness: “I saw the Spirit descend from heaven like a dove, and it remained on him. 33 I myself did not know him, but he who sent me to baptize with water said to me, ‘He on whom you see the Spirit descend and remain, this is he who baptizes with the Holy Spirit.’ 34 And I have seen and have borne witness that this is the Son of God.”
(John 1 ESV)
Each part is consists of an identification and a corresponding action. The combination has been arranged chiastically forming a complete witness from the Baptist about Jesus:
The Lamb of the God [ὁ ἀμνὸς τοῦ θεοῦ]
who takes away the sin of the world
the Spirit descends and abides [τὸ πνεῦμα καταβαῖνον καὶ μένον]
baptizes with Holy Spirit [ὁ βαπτίζων ἐν πνεύματι ἁγίῳ]
The Son of the God [ὁ υἱὸς τοῦ θεοῦ]
One of the main points is the Spirit both descends and abides. Taken literally, John is making the point all of Jesus' ministry after His baptism was done with the Spirit who abides, or remains with Jesus. The Spirit, τὸ πνεῦμα is not called holy, ἅγιος, but the use of the article, the Spirit can be taken as cataphoric. The Spirit is πνεύματι ἁγίῳ.
The phrase Holy Spirit is found three times in John's Gospel:
I myself did not know him, but he who sent me to baptize with water said to me, ‘He on whom you see the Spirit descend and remain, this is he who baptizes with the Holy Spirit.’ (1:33)
But the Helper, the Holy Spirit, whom the Father will send in my name, he will teach you all things and bring to your remembrance all that I have said to you. (14:26)
And when he had said this, he breathed on them and said to them, “Receive the Holy Spirit. (20:22)
English translations have the in all three but the article is only present in the second:
1:33 πνεύματι ἁγίῳ - Spirit Holy
14:26 τὸ πνεῦμα τὸ ἅγιον - the Spirit the Holy
20:22 πνεῦμα ἅγιον - Spirit Holy
When the presence of the article is given weight in these three, the second is unique and the first and last complement one another. The Baptist says Jesus will baptize with (the) Spirit Holy; after the resurrection Jesus breathes on the disciples telling them receive (the) Spirit Holy.
Adding what Jesus told the disciples at the Last Supper speaks of another action: the Father will send the Spirit the Holy in Jesus' name. This will also come with an action, the Spirit the Holy will teach the disciples all things and bring to their remembrance all that Jesus said to them.
At the baptism of Jesus John saw the Spirit, which was identified simply as the Spirit, which remained on Jesus. There is no mention of receiving the Holy Spirit as the disciples will experience after the crucifixion and resurrection.
The words of the Baptist were fulfilled when Jesus breathed on them and the disciples received the Holy Spirit. I would conclude the words Jesus spoke at the Last Supper were fulfilled on the day of Pentecost. So the disciples experienced an internal receiving of the Holy Spirit from Jesus and external receiving on Pentecost, who the Father sent in Jesus' name.
A meaning of ἅγιος is to be separate. The omission in 1:33, the Spirit descend from heaven like a dove, and it remained on him is a Trinitarian construction. Jesus is equal with the Father, so the Spirit is not separate despite being on Jesus. Likewise, the Spirit is also God and so is not separate, despite being away from the Father. On the other hand, once Jesus returns to the Father, the Spirit with which baptism occurs is described using the word ἅγιος. This too is a Trinitarian construction. The Holy Spirit is separate from the Father and the Son while it is with the children of God on the earth.
Finally, there is no corresponding event where, like the disciples, Jesus receives the Holy Spirit. I would conclude the omission confirms the co-equal nature of Son and Spirit. That is, there is no need for the Son of God to receive the Holy Spirit internally. All that is necessary is for the Spirit to abide externally on Jesus so John may identify the Lamb of God who will take away the sin of the world who is the one who baptizes with the Holy Spirit and who is the Son of God.