Yes, He did come upon them before baptism.
This is also true on the day of Pentecost in Acts 2 when He filled the apostles and appeared as 'tongues of fire' resting on each of them.
Why do you suppose only on 2 recorded occasions did the Spirit fill men prior to baptism?
It is important to note that these 2 occasions were major departures from prior theology. The day of Pentecost was the establishment of the Lord's church, closing and fulfilling the covenant age of Moses.
Cornelius represented salvation coming to the Gentiles and revealing the hidden mystery God had intended from the beginning.
I think it is instructive that in Acts 10 Peter and the other Jewish believers remark: received the Holy Spirit just as we did
46 For they were hearing them speaking with tongues and exalting God. Then Peter answered, 47 “Surely no one can refuse the water for these to be baptized who have received the Holy Spirit just as we did, can he?” 48 And he ordered them to be baptized in the name of Jesus Christ. Then they asked him to stay on for a few days.
God moved in such a compelling way that Peter realized that his vision of the sheet being lowered with unclean animals coupled with seeing these Gentiles be immersed in the Holy Spirit in a clearly unique way Just as they had was God's clear stamp of approval for the Gentiles to be part of the kingdom.
So, was Cornelius saved when he believed and was filled with the Holy Spirit, or later when he got wet?
Yes.
These things are so tightly tied together they are inseparable. 'Later' was clearly the same hour. As I have repeatedly stated, the getting wet part is not the focus. Baptism is part of the same submission process that believing is, and baptism takes less 'effort' on your part than believing does, yet no one balks at the notion one must believe.
Peter and Paul both commanded that baptism in the name of the Lord Jesus was a requirement. Jesus told Nicodemus in John 3 that to enter the kingdom one must be born of water and the Spirit, harmonizing the two elements of baptism.
In the case of Acts 19, it was for receiving the Holy Spirit, as is normal for most believers. In the case of Cornelius, it was because they had received the Spirit in a remarkable way.
This also shows that we can not put God in a box. He is Sovereign and He will save who He wants, how and when He wants.
The real question boils down to: will we submit to God's clear instruction?
So, was Cornelius saved when he believed and was filled with the Holy Spirit, or later when he got wet?
“Yes.”
Your answer is ambiguous. So, which was it.
When he was filled with the Holy Spirit?
OR
when he got wet.