One of my friends mentioned to me the New Covenant in connection with Acts 2, which I thought was a little odd, as I can't find a mention of the New Covenant in Acts 2. He tried to connect them via Joel 2:19–32, which I thought was more odd, because I can't find a mention of the New Covenant in Joel 2. But after some thought, I admit there is something there that's worth considering.
It's a fact that the Lord frequently fulfills a prophecy partially before fulfilling it ultimately. We don't have to look hard to find that sort of thing in Matthew's gospel, for example. A great deal of the statements Matthew makes about prophecy seem to be based on quotes wrested from their context. One reason for that (there are other considerations too) is that God sees ultimate fulfillment in Christ. For example, when Isaiah says, "Behold, the virgin shall conceive and shall bring forth a son, and call his name Immanuel" (Isaiah 7:10–25), our minds go immediately to Christ. And rightly so (Matthew 1:23). But in the context where Isaiah actually spoke that prophecy, he was promising deliverance from Syria to Ahaz. If we read on to the next verses (Isaiah 7:14–16), the promise is that a virgin would conceive, have a son, and before her son would be old enough to develop a discerning palate, God would destroy the kings of Syria. So God promises deliverance in the time it takes a young woman to marry, give birth to her first child, wean him, and he develops a taste for food. That's a really weird way to specify a time-frame, but God does it with a purpose: He is looking forward to ultimate deliverance from the ultimate enemy, and that will come from the Son of God, who would be born of a virgin.
So in Acts 2, the coming of the Holy Spirit is identified by Peter (Acts 2:14–21) as the pouring out in Joel 2:28–32. A lot of Christians read Peter's words here and declare that God has fulfilled the prophecy in Joel 2, it's now in the past. I disagree, because Joel clearly promises things that didn't happen in Acts 2. We have no record that the younger people prophesied while the older people dreamed dreams. As Watchman Nee points out, between the Holy Spirit's coming and Peter's sermon, we know for a fact no one had dreamed dreams, because none of them had been sleeping! Peter goes on to mention the signs and wonders Joel promised (Joel 2:30–31; Acts 2:19–20), which is odd considering none of these things are recorded in Acts 2, nor anywhere else in Acts.
Peter wasn't claiming that the pouring out of the Holy Spirit in Acts 2 was the fulfillment of the prophecies in Joel 2. Peter was claiming that what they saw in Acts 2 was a pouring out of the Holy Spirit, and they should have known a pouring out of the Holy Spirit was coming, because it's foretold in Joel 2. This is subtly different: the pouring out of the Holy Spirit is coming, I have no doubt. Acts 2 wasn't the ultimate fulfillment of that prophecy, but it was a partial fulfillment. Just like the child born in Ahaz's time was a fulfillment of Isaiah's prophecy, but a bigger fulfillment was coming; so the pouring out of the Holy Spirit on the day of Pentecost was a fulfillment of Joel's prophecy, but a more complete fulfillment is coming.
Notice this same theme is picked up in Romans 11:25–27. What God has done since Christ was rejected (see especially Acts 3:19–26) is build something new on the earth that gives a glimpse into what things would have been, had Israel repented. And Romans tells us, it's with the intention that Israel would see what God has done in the church, and be moved to jealousy.
And notice, Romans 11:27 mentions the Covenant God will make with Israel "after I take away their sins."
So here's another place the church and the New Covenant meet. Again, I don't believe that the church benefits under the New Covenant, but I absolutely believe that God is using us to demonstrate publicly His grace, so that Israel would be moved to repentance. God is showing how He would treat them, in how He treats us. And part of that promised renewed relationship with Israel is the New Covenant.
I should probably also mention... I don't believe the church is temporary, but I am certain the church is playing roles right now on earth that are temporary. What I mean by that is, our relationship to Christ as Bride isn't going to end, but our place on this earth as the house of God will absolutely come to an end. We will eventually be replaced by Israel in that sense. And that's a good thing! We should look forward to that, because it'll be part of the public vindication of the Lord Jesus Christ. I mentioned before, I don't believe our role as the Body of Christ will last beyond our time here (Christ already has a body in Heaven, our place is to be His Body on earth). I don't believe our role as the habitation of God through the Spirit will last beyond our time here. But we absolutely will be "forever with the Lord" (1 Thessalonians 4:17).