Psalms are not so much direct predictions of the future, but they contain promises which are not entirely fulfilled in the present so look forward to the future. (Sometimes these are refered to as types, and the fulfilments as antitypes.) The OT type promises more than the present, and the NT antitype fulfills the divine purpose implicit in the earlier event. Often Jesus is taken as being the antitype of descriptions in the psalms. That is, he fulfils the description of a psalm even though it was a personal testimony, not a ‘Thus says the Lord: in the latter days…’ prophecy. [The most obvious one is Ps.22.] So Jesus is the perfect man linking Eden to Hebrews in Psalm 8, he’s the perfect priest linking Genesis 14 to Hebrews (5 & 6) in Psalm 110, he’s the perfect King beyond David’s experience, the perfect Worshipper beyond the faithful Israelite and the perfect Object of worship, etc. This isn’t usually a straight ‘fit the peg in the hole’ fulfilment: the antitype is usually greater than the type. Jesus more than fulfils the psalms in every way. E.g. He suffers for righteousness’ sake, as the king over God’s people, like David, but he has no need for confession, unlike David: David is a shadow of the King to come. So we’re not looking just for sentences-which-only-Jesus-fulfils sprinkled randomly through the psalms. We’ll be thinking, What did this mean at the time – in what way was it the psalmist’s experience / hope / lament / prayer / praise? And how does Jesus fulfil this even more?
You are not the hero
The hero of the psalm, the more-than-fulfilment, the antitype, is Jesus – not us! Now, once the NT applies it to Jesus, then because we are in Him, and co-heirs of his righteousness, then some of it applies to us too (E.g. Ps.2, Rev.12, Rev.2.26-27). But it doesn’t apply directly to us without going through Jesus – in fact, it didn’t even apply to the psalmist without going through Jesus, the Messiah, but they didn’t say that so explicitly. (There are some hints of awareness e.g. in Ps.40.)
I'm not entirely happy with how I phrased everything then, and I'm still thinking through links between Christology and the doctrine of revelation as it applies to the psalms. Comments on that specifically very welcome.
But I sometimes think that we're so caught up with what the question is that we forget we're all agreeing on the beautiful answer that every child in Sunday School can give us: Jesus!