Beginning (arxamenos). Aorist middle participle of archō, agreeing (nominative) with ho kurios Iēsous (the Lord Jesus). The ministry of Jesus began with the ministry of John. Strictly speaking arxamenos should be the accusative and agree with martura (witness) in Act 1:22, 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 (martura tēs anastaseōs autou sun hēmin). 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.