"Spiritual person" is a person filled with the Holy Spirit, who enjoys the presence of the Holy Spirit in himself and is led by the same Spirit towards cognition of the divine actions. One of those divine actions is judgment, for God-the Father judges (for if He does not, how could Jesus ask Him not to judge His murderers /Luke 23:35/?), and God-the Son judges with a true judgment (John 8:16), and God-the Holy Spirit judges (John 16:8), for the judgment is the sole prerogative of the Trinity only and not of any even highest of angels or creatures.
However, God through His Son and Spirit bestowed upon humans also the grace of judgment, and the reward of those who follow Jesus to the end is that they will be advanced to a divine status of "judging 12 tribes of Israel" (Matthew 19:28), and also Paul admits that the Spirit-bearing believers will be in a divine status of "judging even angels" (1 Cor. 6:3). But humans will possess this divine quality by participation, not essentially, as the Trinitarian Persons of the Father and the Son and the Holy Spirit, which is a crucial difference. Just like the divine quality to forgive is in the Trinity essentially, but to humans God gives this quality by participation, that is to say, by grace of the Spirit (Matthew 18:18).
This is how the passage should be understood: Spirit-ual man, that is to say, a man possessing Holy Spirit is authorised to judge every thing and every person, because in virtue of being led by Spirit, and in virtue of being a co-actor with the Spirit, he judges not humanly but divinely. Human judgment of a natural man is sinful and wrong, because it is limited, cruel, devoid of love, distorted by passions and egotistic drives that beset all natural men. But a man who acts in Spirit is devoid of those passions and is a clean mirror that reflects a clear divine light to all. Such a man will necessarily judge and estimate all things, for God does so and he being endowed by God-Spirit's presence will do the same.
Moreover, such a man cannot be judged by anybody, for who can judge the actions of Spirit that he transmits by his spiritual life of co-action with the Spirit? The very principle of judgment is provided by Spirit's gift of understanding of what is right and what is wrong, and how can anybody judge a Spirit-led man, by what principle? If this principle is not from Holy Spirit, then it is stupid to base judgment on this non-Spiritual principle, but if it is from Holy Spirit, then this judging person will also be Spiritual and so will have no reason to judge another Spiritual person's actions, for Spirit does not oppose Himself. Thus, natural men are illegitimate to judge the Spiritual men, for the very principle of just judgment is provided by the presence of Spirit in us, of which presence a natural man is devoid.