The first thing I notice is that Paul clarifies himself in the final verse, which is a quote from Isaiah 40:12-14 (NJPS):
Who measured the waters with the hollow of His hand,
And gauged the skies with a span,
And meted earth’s dust with a measure,
And weighed the mountains with a scale
And the hills with a balance?
Who has plumbed the mind of the Lord,
What man could tell Him His plan?
Whom did He consult, and who taught Him,
Guided Him in the way of right?
Who guided Him in knowledge
And showed Him the path of wisdom?
Second, the word anakrino <350>, pointed out in the question, can mean:
1) examine or judge
1a) to investigate, examine, enquire into, scrutinise, sift, question
1a1) specifically in a forensic sense of a judge to hold an
1a2) to interrogate, examine the accused or witnesses
1b) to judge of, estimate, determine (the excellence or defects of
any person or thing
So it could be translated "examine", which doesn't carry the same connotation that "judge" does. In fact, the Isaiah reference (both the quick quote and the broad context of the quotation) makes quite clear that Paul is not talking about spiritual people being in judgement over others (which is sense 1b above). Rather, the spiritual person has greater insight into the world around us, yet is not understood by people who lack "the mind of Christ" as Paul says at the end of verse 16.
Paul goes into much more detail in chapter 15. In particular:
So is it with the resurrection of the dead. What is sown is perishable; what is raised is imperishable. It is sown in dishonor; it is raised in glory. It is sown in weakness; it is raised in power. It is sown a natural body; it is raised a spiritual body. If there is a natural body, there is also a spiritual body. Thus it is written, “The first man Adam became a living being”; the last Adam became a life-giving spirit. But it is not the spiritual that is first but the natural, and then the spiritual. The first man was from the earth, a man of dust; the second man is from heaven. As was the man of dust, so also are those who are of the dust, and as is the man of heaven, so also are those who are of heaven. Just as we have borne the image of the man of dust, we shall also bear the image of the man of heaven.—1st Corinthians 15:42-49 (ESV)
Paul makes clear in other places (1st Corinthians 13:12, Romans 7:21-25, etc.), that we don't yet have resurrection bodies, but we do have God's Spirit in the meantime. So for now, the spiritual person can see things that the natural person (who has a natural soul) doesn't understand. In fact, the natural person can't even understand the spiritual person at all.
The NET Bible supports that reading:
The one who is spiritual discerns1 all things, yet he himself is understood2 by no one.
The notes are:
1 tn Or “evaluates.”
2 tn Or “is evaluated” (i.e., “is subject to evaluation”); Grk “he himself is discerned,” that is, the person without the Spirit does not understand the person with the Spirit, particularly in relation to the life of faith.