Christ and the Holy Spirit are both “apostolic” because both have been sent by the Father to the World. It is not only repeated in the Scripture on numerous occasions how Christ has been sent by the Father, and the Spirit sent through Christ from the Father, but it also has been recorded explicitly that Christ is “the apostle… of our confession” (Heb 3:1).
As Christ was sent from God, so Christ Himself chose and sent His apostles. “As the Father has sent me, even so I send you… receive ye the Holy Spirit,” the risen Christ says to His disciples. Thus, the apostles go out to the world, becoming the first foundation of the Christian Church.
In this sense, then, the Church is called apostolic: first, as it is built upon Christ and the Holy Spirit sent from God and upon those apostles who were sent by Christ, filled with the Holy Spirit; and secondly, as the Church in its earthly members is itself sent by God to bear witness to His Kingdom, to keep His word and to do His will and His works in this world.