This verse does not remotely provide what is needed to justify solo singing. First, the passage does not mention singing. A psalm can be read, or quoted, as easily as it can be sung. Further, a psalm could be presented to the congregation for learning without a solo being performed. If a verse does not explicitly state a truth, or at least necessarily imply it, no speculation should be made by which to justify some coveted practice.
Imagination is a poor base upon which to construct an argument.
In this connection we would make passing reference to the case of the disciples as they assembled in Acts 4, celebrating the release of Peter and John from prison. Moreover, if a psalm were sung under the influence of the Spirit for instructive purposes, that would have no bearing upon what the church is allowed to do today. Third, it appears fairly obvious that Paul, in this context, is attempting to correct an abuse. If such is the case, this verse is hardly one to be citing in support of the chorus-solo system. The truth is, the New Testament is void of authority for solo and choir music in church worship.