How does Western Christian theology explain the doctrine of the Filioque within the context of the Old Testament?

If I understand the question, it seems to be along the lines of "How can Jesus be the one who sends the Holy Spirit, when the Holy Spirit is sent in the Old Testament?"

There's two threads that help answer this. The first is that Jesus is there in the Old Testament. The second is that the Holy Spirit was only given to a select few in the Old Testament, not all the people.

Regarding the first thread or point, Jesus is the incarnate Word of God. He is the same Word through whom God spoke everything into existence in Genesis 1; John makes this point in the opening of his Gospel. Jesus is also the "angel of the Lord" who appears throughout the Old Testament. The "angel of the Lord" is the Lord Himself, the pre-incarnate Jesus Christ; this is seen, for instance, in Genesis 22, Exodus 3, and the visit of the three men to Abraham in Genesis 18, one of whom Abraham calls "the Lord." So, Jesus was there in the first act of creation and is with his people throughout the Old Testament.

Regarding the second thread or point, in the Old Testament the Holy Spirit is only given to certain people, mainly the prophets (although the kings such as Saul, David, and Solomon also had the Holy Spirit, see for instance 1 Samuel 16:14 when the Spirit departs from Saul). In fact, in Numbers 11:29, after the Lord takes some of the Spirit He had given Moses and gives the Spirit to the elders of Israel, Moses says, "Would that all the LORD's people were prophets, that the LORD would put his Spirit on them!" Then, in Joel 2:29, the Lord promises that He will one day pour out His Spirit upon "al flesh." Peter connects this promise with Pentecost in Acts 2:17-18; this is, in the New Testament Church the Lord is fulfilling His promise to pour out His Spirit upon "all flesh."

Thus, owing to Christ's ascension back into heaven to take his seat at the right hand of the Father, the Holy Spirit is poured out on all believers, not just the select few as in the Old Testament. This is visually seen in Revelation 5 when John sees Christ, the Lamb "with seven horns and with seven eyes, which are the seven spirits of God sent out into all the earth" (Rev. 5:6). These "seven spirits" symbolize the Holy Spirit who Christ sends in fulfillment of the longing of Moses, the promise of the Lord through Joel, and Christ's own promises (cf. John 14:26, 15:26).

I think the Filioque is just trying to get across the point that the Holy Spirit is sent to all believers due to Christ's ascension as well as stress the essential divine unity of Father, Son, and Holy Spirit. They are "one God in three persons." The Son does the Father's will and together they send the Holy Spirit to witness to Christ and bring people to faith.

A particularly pertinent passage of scripture is found in 1 Peter 1:11 - in surrounding context:

Concerning this salvation, the prophets who prophesied about the grace that was to be yours searched and inquired carefully, inquiring what person or time the Spirit of Christ in them was indicating when he predicted the sufferings of Christ and the subsequent glories. It was revealed to them that they were serving not themselves but you, in the things that have now been announced to you through those who preached the good news to you by the Holy Spirit sent from heaven, things into which angels long to look. - 1 Peter 1:10-12 ESV (emphasis added)

Contemplation of how the Holy Spirit could be particularly denoted as the Spirit of Christ in His operation upon the Old Testament prophets leads naturally to a "Proceeding from (or more correctly, through) the Son" perspective.