Did early Christians believe the world was about to end?

Neither of the two verses in the question have Jesus (or anybody else) using the phrase, "the end of the world". People today have all sorts of ideas about what Christians should believe about the world ending (either literally or symbolically), but if we can just stick to what Jesus actually said, the answer might become clear.

First, the Matthew 16:28 statement. Jesus promised that some of his disciples would not die before they saw him "coming in his Kingdom". The very next verse tells us that six days later, he took three of them up a mountain where they saw him transfigured into such glory as they had never seen before. They witnessed Moses and Elijah talking with this glorified Christ. Then God spoke from a bright, overshadowing cloud, saying, "This is my beloved Son, in whom I am well pleased; hear ye him." Then the vision vanished and only Jesus was there with them.

Surely that was the fulfillment of Jesus' promise six days earlier? And also consider 95 A.D. when the aged apostle John had visions of the resurrected glorified Christ in heaven. The descriptions of Christ are of this majestic King of heaven who shares the Father's throne (Revelation 5:5-14 & 19:11-16 & 20:11-13 & 22:1-3). He is crowned, he rules, he judges all the dead whom he has raised, all the powers of evil are defeated by him, and a new heaven and a new earth comes into being. If that was not John seeing Christ coming in his Kingdom power and glory, what was?

Second, the Mark 13:30 statement about the generation that would "not pass" until "all these things be done" (A.V.). What things? From verse 1 till verse 23 Jesus was answering the disciples' question as to when the great Jerusalem temple buildings would be ruined. He built up a prophetic picture of increasingly awful tribulation for his followers (always addressed by him in this chapter as "you"). But he warned them not to heed false Christs claiming they where Christ returned. He warned them that false prophets would arise to try to deceive them. That takes us to the end of verse 22. Now comes a clear division, or section in Jesus' prophecy:

"But take ye heed: behold I have foretold you all things. But in those days after that tribulation, the sun shall be darkened, and the moon shall not give her light, and the stars of heaven shall fall, and the powers that are in heaven shall be shaken. And then they shall see the Son of man coming in the clouds with great power and glory. And then he shall send his angels, and shall gather together his elect from the four winds, from the uttermost part of the earth to the uttermost part of heaven." Mark 13:23-27 A.V.

Notice how Jesus begins to speak of "they", as opposed to his followers, who he always calls "you"? That's because his followers are all safely and suddenly gathered to him at that time, which is the time "after that tribulation" - the one his followers have gone through. Before they see Jesus in glory, in his Kingdom power descending from heaven to take them safely away, they have gone through great tribulation. Now, has that gathering of the elect to be with Jesus at his return happened yet? No. Are Christians today still going through increasingly great tribulation? Yes. Are Christians in this generation still eagerly awaiting Christ's sudden return as they were in the first century? Yes. So, the answer to the question is that every generation of Christ's followers have longingly awaited seeing him return to Earth in his Kingdom glory, from when Jesus told them to look out for his return, until he actually does return - which has not yet happened.

Yet it is clearly true that those first century Christians did experience great tribulation leading up to the literal fulfilment of the earthly destruction of the Jerusalem temple in A.D. 70. They witnessed that destructions, being protected from it by heeding Jesus' warning to flee once they saw armies encamped around Jerusalem. When those Roman armies suddenly stopped their siege and went away, that was when they fled. When the armies later returned, all who had not heeded Jesus' warning either died horribly or were taken off as slaves. Not the Christians who believed Jesus' prophecy.

The first part of his prophecy having been fulfilled within their generation, the second part then started, which part continues to this day. As explained here:

"The temple would fall amidst dreadful affliction. However this should pale into insignificance compared to that attending the fall of Christendom in the coming days of unprecedented tribulation. Nevertheless the elect are forewarned of the certainty that despite these signs the end is not yet, not whilst the earth trembles, but when the heavens change, then the end is nigh. During the former times tribulation will trouble the whole world, to mount increasingly as the new - heavenly - portents loom nearer and nearer, Mk. 13:24-27.

Intensifying tribulation on earth; furious persecutions in the world and from its hostile religion; dreadful global visitations. But the end is not yet. The elect must endure all with patience, expecting no relief, knowing that this is but the herald of new signs in a different and heavenly sphere. 'The sun shall be darkened, and the moon shall not give her light. And the stars of heaven shall fall, and the powers that are in heaven shall be shaken.' Then; mark that, then - not till then - 'Then shall they see the Son of man coming in the clouds with great power and glory.' Mk. 13:26

...The elect being raised from the dead, delivered out of the world, gathered to Christ in glory, nothing remains for the world, and for this world's religion, but the fiery deluge, the dissolution of all things, the resurrection of damnation, the final judgment, and the vengeance of eternal fire." Mark, John Metcalfe, pp. 183-5, John Metcalfe Publishing Trust, 1996

Conclusion Summarized: The first century Christians were looking for the destruction of the Jerusalem temple as a major sign indicating Christ's return would happen. But after its destruction in A.D. 70, and Christ still had not appeared by A.D. 95, John had visions of Christ in kingly kingdom power, revealing far more details about events leading up to his sudden return. That fortified the Christians through the next century where, in every succeeding generation they held on to that prophetic word to keep them faithful through increasingly dreadful tribulations - to this very day.

Every generation of Christians has been told enough to keep them looking in faith for Christ's spectacular return to Earth, obediently carrying out God's will for them, and for the honour of his name. Christ told us all that he would go away to a 'far country' [heaven] for a long time, then return to hold everyone to account [the Last Day]. That Day of Resurrection and Judgment is nearly 2,000 years nearer now than it was when Jesus spoke of "that tribulation" (for the early church) and the great tribulation just before his return.

Historical-critical scholars such as Albert Schweitzer (1875-1965) believed Jesus was a failed prophet.

"...saying that no one, including himself, knew the exact time of his return, but it would be before the end of the end of the first generation of followers." wikipedia

I didn't notice a more traditional understanding of this concept in your link. It is called "the continual coming of Christ in the church." It was taught by those like Augustine (354-430), Tyconius (about 380), Origen (184-253), and though it is generally misunderstood, Justin Martyr (100-165). Augustine explained it this way.

"--whether to that coming of the Saviour which continually occurs in His Church, that is, in His members, in which comes little by little, and piece by piece, since the whole Church is His body, or to the destruction of the earthly Jerusalem." St. Augustine of Hippo: The City of God Book XX, ch 5

Earlier Origen talked about "this generation."

Origen: The uninstructed refer the words to the destruction of Jerusalem, and suppose them to have been said of that generation which saw Christ’s death, that it should not pass away before the city should be destroyed. But I doubt that they would succeed in thus expounding every word from that, “one stone shall not be left upon another,” to that, “it is even at the door;” in some perhaps they would succeed, in others not altogether. Origen: Yet shall the generation of the Church survive the whole of this world, that it may inherit the world to come, yet it shall not pass away until all these things have come to pass. But when all these shall have been fulfilled, then not the earth only but the heavens also shall pass away; that is, not only the men whose life is earthly, and who are therefore called the earth, but also they whose conversation is in heaven, and who are therefore called the heaven; these “shall pass away” to things to come, that they may come to better things. But the words spoken by the Saviour shall not pass away, because they effect and shall ever effect their purpose; but the perfect and they that admit no further improvement, passing through what they are, come to that which they are not; and this is that, “My words shall not pass away.” And perhaps the words of Moses and the Prophets have passed away, because all that they prophesied has been fulfilled; but the words of Christ are always complete, daily fulfilling and to be fulfilled in the saints. Or perhaps we ought not to say that the words of Moses and the Prophets are once for all fulfilled; seeing they also are the words of the Son of God, and are fulfilled continually. Catena Aurea

Referred to as chiliasm, Justin Martyr referred to a thousand years in Jerusalem and compared it to both the carnal and spiritual life of Adam.

I and others, who are right-minded Christians on all points, are assured that there will be a resurrection of the dead, and a thousand years in Jerusalem, which will then be built, adorned, and enlarged, [as] the prophets Ezekiel and Isaiah and others declare. (Dialogue with Trypho, ch LXXX) For as Adam was told that in the day he ate of the tree he would die, we know that he did not complete a thousand years. (Dialogue with Trypho, ch LXXXI)

The concept of spiritual life is best compared to New Jerusalem rather than the earthly Jerusalem.

(Heb 4:3)  For we which have believed do enter into rest, as he said, As I have sworn in my wrath, if they shall enter into my rest: although the works were finished from the foundation of the world. (Heb 4:4)  For he spake in a certain place of the seventh day on this wise, And God did rest the seventh day from all his works. (Heb 12:22)  But ye are come unto mount Sion, and unto the city of the living God, the heavenly Jerusalem, and to an innumerable company of angels, (Heb 12:23)  To the general assembly and church of the firstborn, which are written in heaven, and to God the Judge of all, and to the spirits of just men made perfect, (Heb 12:24)  And to Jesus the mediator of the new covenant, and to the blood of sprinkling, that speaketh better things than that of Abel.

This can be understood as the spiritual light in our hearts.

2Pe 1:19  We have also a more sure word of prophecy; whereunto ye do well that ye take heed, as unto a light that shineth in a dark place, until the day dawn, and the day star arise in your hearts: 2Co 4:4  In whom the god of this world hath blinded the minds of them which believe not, lest the light of the glorious gospel of Christ, who is the image of God, should shine unto them.  2Co 4:5  For we preach not ourselves, but Christ Jesus the Lord; and ourselves your servants for Jesus' sake.  2Co 4:6  For God, who commanded the light to shine out of darkness, hath shined in our hearts, to give the light of the knowledge of the glory of God in the face of Jesus Christ.

So the coming Jesus sometimes referred to was present in His generation.

The text you link to, Matthew 16:28, is so very closely linked to the next verse, '... and after six days' that its natural interpretation is that the those present actually saw the coming of the Son of man on the mount of transfiguration and that Jesus' words relate to that experience, not to the end of the world.

But it is evident that the early church viewed the return of Christ as immediate. That is to say, as imminent in their own, personal apprehension.

Paul says to Titus :

... the grace of God that bringeth salvation ... looking for that blessed hope and the glorious appearing of the Great God and our Saviour, Jesus Christ. [Titus 2: 11, 13 KJV.]

The urgency and immediacy of the expectation was a hope and a desire. Faith looked in hope for the coming of the Lord Jesus. Love longed to see him as he really is, either again or for the first time.

The women who went to the tomb went there in love, to anoint the body of Jesus. Mary was bereft without him, longing just to find his deceased body, frustrated because someone appeared to have stolen him, John 20:15.

After his ascension, the disciples had to be exhorted to activity by no less than angels as they were standing around looking at the spot in the sky into which Jesus had disappeared, Acts 1:10 and 11.

Paul had to exhort some of the Thessalonians that if a man will not work neither let him eat as it appears some had given up working because of the imminence of the Lord returning and the world ending. Paul had to remonstrate with them that it was irresponsible to respond in that way, II Thessalonians 3 : 6-15.

At the end of his life, Paul exhorts Timothy :

Henceforth there is laid up for me a crown of righteousness which the Lord, the righteous judge, shall give me at that day : and not to me only, but unto all them also that love his appearing, [II Timothy 4: 8 KJV.]

It was love that prompted the early church to look for the appearing of the Lord.

Towards the end of that era, as scripture closed, John the apostle, aged then between ninety and more than an hundred years old - at the very end of his Revelation and at the close of the first century, writes :

Surely I come quickly. Amen. Even so, come quickly, Lord Jesus. [Revelation 22:20 KJV.]

Yes, I see that "early Christians viewed the end of the world as imminent". And I believe that the bible itself as a "source" demonstrates that views about the apocalypse did not change at all during the first century.

For further generations, faith and love and hope should be unchanged, despite the passage of time, despite the passing generations. For with the Lord, a day is as a thousand years, and just a couple of days have passed since he ascended. And the Lord is not slack concerning his promise, II Peter 3:8.

Faith and hope and love still gaze upwards, imminently expecting the return of the Lord Jesus.