A.T. Robertson Commentary Acts 1:22

"beginning from the baptism of John, unto the day that he was received up from us, of these must one become a witness with us of his resurrection." — Acts 1:22 (ASV)

Beginning (αρξαμενος). Aorist middle participle of αρχω, agreeing (nominative) with ο κυριος Ιησους (the Lord Jesus). The ministry of Jesus began with the ministry of John. Strictly speaking αρξαμενος should be the accusative and agree with μαρτυρα (witness) in verse 22, but the construction is a bit free. The ministry of Jesus began with the baptism of John and lasted until the Ascension.

A witness with us of his resurrection (μαρτυρα της αναστασεως αυτου συν ημιν). This Peter considers the essential thing in a successor to Judas. The one chosen should be a personal witness who can speak from his own experience of the ministry, resurrection, and ascension of the Lord Jesus. One can easily see that this qualification will soon put an end to those who bear such personal testimony.