Was the Holy Spirit already in existence during the Old Testament or only
after the New Testament? What was Holy Spirit doing in the beginning?

The Holy Spirit, being God, existed before the creation of the world. He
was not created, but has always existed, just like the Father and the Son.
As proof of this, one can look right back in Genesis 1:1, “and the Spirit
of God was hovering over the waters.” Although the Old Testament does not
go into detail about the Father, the Son and the Holy Spirit, there are
hints of all three aspects of the godhead scattered throughout the Old
Testament. You ask what the Holy Spirit was doing at the beginning.
Genesis 1:1 gives one hint. Apparently the Holy Spirit was involved in
creating the world. John 16:8 shows that part of the work of the Holy
Spirit is “to convict the world of guilt in regard to sin and
righteousness and judgment: in regard to sin, because men do not believe
in me; in regard to righteousness, because I am going to the Father…” I
would assume that since mankind have existed on the earth, the Holy Spirit
has been working in this way. There are a number of instances in which one
can see the Holy Spirit working directly in the Old Testament. Examples of
this would include Psalms 51:11, in which David asked God not to remove
his Holy Spirit from him and 1 Sam 19:19-24 in which the Spirit is even
poured out on King Saul.

In summary, there is no other way to view the Bible’s picture of the Holy
Spirit except in the context of him being an eternal part of God. Although
the Holy Spirit is given to those who are in Christ in a special way, he
has always worked to bring people to God.

John Oakes, PhD