Did all disciples of Christ's apostles consider priesthood a sacrament? Is it true that all the disciples of Christ's apostles (including Paul) considered (in their writings of the 1st and 2nd centuries) the priesthood one of the sacraments?
Simple answer, No. No there is no proof of the assertion by that priest. 

The idea of Apostolic succession from which the orthodox imposition of hands as a sacrament finds its purpose is not something that can be historically traced. The laying on of hands is in the bible but not for this serious purpose of defining legitimate authority to administer the sacraments through a chain of succession. 

As there are little writings after the Apostles before Tertullian (c. 160 – c. 225 AD. The best I can argue is by finding the best evidence for the 'imposition of hands' as a holy sacriment to trasmit succession and show it to be no proof at all.  

I will not turn to the scripture as there is no evidence there to support this sacrament. The place where you would most expect to find it is in 1 Timothy Chapter 3, as the qualifications for bishops are listed, but nothing is there. This at once proves the requirement not set out as a  scripturally required rite, but looks look at the argument from history alone.

 Here is a quote by Tertullian making the best case for a kind of succesion from a [wiki] write up:

Tertullian,  In his Prescription Against Heretics, he explicitly challenges heretics to produce evidence of the apostolic succession of their communities.


  Let them produce the original records of their churches; let them unfold the roll of their bishops, running down in due succession from the beginning in such a manner that [that first bishop of theirs] bishop shall be able to show for his ordainer and predecessor some one of the apostles or of apostolic men, — a man, moreover, who continued steadfast with the apostles. For this is the manner in which the apostolic churches transmit their registers: as the church of Smyrna, which records that Polycarp was placed therein by John; as also the church of Rome, which makes Clement to have been ordained in like manner by Peter. In exactly the same way the other churches likewise exhibit (their several worthies), whom, as having been appointed to their episcopal places by apostles, they regard as transmitters of the apostolic seed."


But notice four things. 

 - First, the emphasis is not about laying on of hands. That is a biblical principle, and every church sort of follows that, in prayer and ordination, but its not portrayed as a the-big-thing. The importance is in the fact that the previous Bishop vouchsafed the new one. Its about credibility.
 - Second, every church had a different bishop, passing the torch. There was no super bishop holding all the cards. Even the Apostles did not 'take charge' over churches planted by another Apostle.
 - Third, To trace such succession today would be impractical, for it would be 100,000 times more complex than Christ's genealogies, for any given city. 
 - Fourth, although this would have been a wise practice in the early church to prevent heresy from totally destroying it,  there is actually nothing in the Bible that says it needed to be that way.

Therefore even if one could prove the impossible to prove (due to lack of historical writings), there is nothing in scripture to enforce its continuance, which would be impractical to do anyway.

I do not think God would allow utter silence in his word and in post apostolic history for something as important as this, if there was any truth to it.

Personally I would find it harder to disprove that God prefers blue over purple, because I see more blue in nature. That does not infer that we can't reasonable conclude God does not prefer blue.