Yefeth ben Ali, the Jewish scholar from 10th century gives an explanation of the Book of Isaiah in regard to the suffering of the King Messiah in his commentary of the Torah:

... to regard it as alluding to theThe Messiah, and as opening with a description of his [King Messiah’s] condition in exile, fromthe time of his birth to his accession to the throne: for the prophet [Isaiah] begins byspeaking of his being seated in a position of great honour, and then goes backto relate all that will happen to him during the captivity. He thus gives us tounderstand two things: In the first instance, that the Messiah will only reach his highest degree of honour after long and severe trials; and secondly,that these trials will be sent upon him as a kind of sign, so that, if he findshimself under the yoke of misfortunes whilst remaining pure in his actions, hemay know that he is the desired one…

* S. R. Driver & A. Neubauer (editors). 1969. The Fifty-third Chapter ofIsaiah According to the Jewish Interpreters (2 volumes; New York: Ktav), pp. 19-20.The English translations used here are taken from volume 2. The original texts are involume 1. Cf. Soloff, pp. 107-09.

There is a very important passage in the very old book Pesikta, cited in the treatise Abkath Rokhel (andreprinted in Hulsii Theologia Judaica, where this passage occurs, p. 309):

When Elohim created the world... under the Throne of His kavod, and brought forth the being of the Messiah. He said to him: 'Will you heal and redeem My sons [servants] after 6000 years?' He [King Messiah] answered Him, 'I will.' Then Elohim said to him: 'Will you then also bear the punishment... as it is written, “But he bore our diseases” [Yeshayahu 53:4]. And the Messiah answered Him; 'I WILL JOYFULLY BEAR THEM' (cf. Zohar, 2:212a).
Rabbi Moses Haddarshan states:

Immediately the Messiah, OUT OF LOVE, TOOK UPON HIMSELF ALL THOSE PLAGUES AND SUFFERINGS, as it is written in Yeshayahu 53,“HE WAS ABUSED AND OPPRESSED.”

Regarding the sufferings of Moshiach, we read in the notes of the Steinsaltz Edition of the Talmud:

According to Rashi, Rabbi Yehudah HaNasi and Daniel were suggested as possible Messiahs [by Rav, d. 247, see Sanhedrin 98b], for they were both totally righteous men and they both suffered severely from various afflictions.
* The Talmud, SteinhaltzEdition (New York: Randomhaus, 1999), notes p. 25, 26.

A righteous one suffering on behalf of the whole world is seen in the Zohar:

The children of the world are members one of another. When the Holy One desires to give healing to the world, he smites one just man amongst them, and for his sake heals all the rest. Whence do we learn this? From the saying (Isaiah 53:5),‘He was wounded for our transgressions, bruised for our iniquities…’

* Zohar, Numbers, Pinchus 218a; Driver and Neubauer, Zohar, Numbers, Pinchus 218a (English translation), p. 15.

The seventy sounds of a woman in labor:

At that time, She is opened to give birth to Messiah. This is owing to the pain and distress of the pious, men of good qualities, sages of secrets of the Torah, people of humility and shame, of fear and love, of kindness, men of valor, fearing Elohim, truthful men, who despise bribes, for whom it is a time of poverty. This is what the sages of the Mishnah have explained. In the times in which the son of David will come, a valiant men will go circulating from city to city but will not be shown favor. Those that fear sin will be despised and the wisdom of scribes will be sullied. Truth will be absent...