To answer the question I think we need to need to consider the whole verse and its context.
Our English translations begin with “therefore” (οὖν), which suggests that Peter is drawing a conclusion from the previous verses (vv.18–22), where Peter writes about Christ’s victory over hostile powers through his death and resurrection.
The connection between the two sections appears to be this: Since Christ’s suffering is the pathway to glory, believers should also prepare themselves to suffer, knowing that suffering is the prelude to an eschatological reward. The main point of the verse is that believers are to arm themselves (hoplisasthe) with the willingness to suffer.
Now notice the imperative verb "ὁπλίσασθε", which translates “arm yourselves.” It has obvious military connotations, and in the Bible the Christian life is frequently compared to the life of a warrior (Rom 6:13; 13:12; 2 Cor 6:7; 10:4; Eph 6:11–17; 1 Thess 5:8) This martial language indicates that discipline and grit are needed to live the Christian life, particularly in view of the suffering which believers encounter. Indeed, believers must arm themselves with the “attitude” (ἔννοιαν) that suffering is inevitable.
The first clause in the verse explains the reason the reader should expect to suffer. Christ also “suffered in the flesh.” The wording here seems to refer back to v.18, where both the verb “suffer” (ἔπαθεν) and the noun “flesh” (σαρκὶ) occur as well. This verse reads:
"For Christ also suffered once for sins, the just for the unjust, that He might bring us to God, being put to death in the flesh but made alive by the Spirit” (1 Peter 3:18 NKJV).
This brings us to the answer of your first question. The suffering of Christ is the suffering he endured upon the cross as he was put to death for our sin.
Notice by way of further evidence that in both of these texts Peter links the suffering of Christ to the suffering of his readers. Notice as well that as we read on, Peter acknowledges a distinction: Our sufferings and Christ’s sufferings are not exactly the same. Christ’s suffering here focuses on his death, as in 3:18 and 2:21–24, but Peter does not imply that our suffering will involve a similar death. In 2:21–23 Peter tells us that Christ’s suffering is an example for believers, providing the pattern they should imitate.
“For to this you were called, because Christ also suffered for us, leaving us an example, that you should follow His steps: Who committed no sin, Nor was deceit found in His mouth; who, when He was reviled, did not revile in return; when He suffered, He did not threaten, but committed Himself to Him who judges righteously.”
This brings us to the answer to your third question. "Suffering in the flesh" refers to suffering experienced in this physical dimension of life. It is similar to Christ’s suffering, but it may not be unto death, which suggests to me that believers are expected to suffer for the sake of the gospel just as Jesus did.
The most difficult part of the verse is the last phrase (and this is your second question),
“. . . for he who has suffered in the flesh has ceased from sin.”
Notice again the word “flesh” (σαρκὶ), which preserves the connection between Christ’s suffering and believers' suffering. There is debate, however, regarding the reason for this connection. Three different interpretations are quite possible. (Achtemeier mentions a fourth; namely, that one’s suffering atones for sin. He rejects this as incompatible with Petrine theology (1 Peter, 279).
First, Michaels (1 Peter, 226–29) and Hillyer (1 and 2 Peter, Jude, 120) (cf. Richard, in 1 Peter, Jude, and 2 Peter, 167–68) suggest the one who suffered could be identified as Jesus Christ. Similarly Bechtler says the verse is ambiguous and both Christ and believers are in view (Following in His Steps, 196–98).
The objection to this view is that Jesus never sinned (cf. 2:22; 3:18), so how could it be said that he had ceased from sin? This view does, however, deal with the problem of human beings becoming sinless in this life. The need to posit Christ as the subject can be eliminated if we show that there are plausible ways of speaking of Christians as ceasing from sin and yet not being sinless. Both of the following interpretations fit this requirement.
Second, according to Calvin, (Catholic Epistles, 121), Beare, (First Peter, 153), Dalton (Proclamation to Spirits, 244–48), Kelly (Peter and Jude, 168–69), and Cranfield, (I & II Peter and Jude, 108), he who suffers in the flesh is a Christian, but his or her suffering should be understood in terms similar to Rom 6:7,
“Anyone who has died has been freed from sin.”
In Romans, chapter 6, believers died to sin by being baptized with Christ in death. Moreover, the dominion of sin has been broken in the lives of those who have died with Christ.
The advantage of this interpretation is that it agrees with Paul and sensibly explains how believers cease from sin. The systematic theologian finds this interpretation attractive. That interpretation should be rejected, however. The context of both Romans and 1 Peter are quite different. In Romans 6, the believer “dies” with Christ, but no such language is used in 1 Peter. Indeed, as we have already seen, the word suffered in v.1 should be equated with dying. As Elliott argues, Paul spoke metaphorically of dying with Christ, whereas Peter had actual suffering in mind (in 1 Peter, 716). The notion here is not that believers have died with Christ but that they should follow Christ in their daily lives by consenting to suffering. Furthermore, Peter, unlike Paul, did not use the word sin (ἁμαρτίας) to designate a power. The word sin in Peter refers to acts of sin (cf. 2:22 and 24; 4:1 and 8).
The third interpretation seems to be the most persuasive. This one is presented by men like Grudem (1 Peter, 167), Schweizer (“1. Petrus 4, 6,” 84), Omanson (“Suffering,” 445–460), Achtemeier (1 Peter, 280), and Bigg (Epistles of Peter and Jude, 167). They all posit that “he who has suffered” refers to believers and relates back to the imperative to prepare themselves for suffering.
Peter explained why they should prepare themselves to suffer; namely, their commitment to suffer serves as evidence they have made a clean break with the life of sin. Peter is not suggesting that believers who suffer have attained sinless perfection. Nor is he suggesting that suffering itself has some power to break the inclination to sin. Rather, Peter is emphasizing that those who commit themselves to suffer for the sake of the gospel demonstrate they have triumphed over sin. In other words, their commitment to suffer reveals a passion for a new way of life, a life that is not yet perfect but is nevertheless remarkably different from the lives of unbelievers in the Greco-Roman world.