According to Catholicism, is are there any teachings or traditions that may shed light on who was the host of the house where Jesus celebrated his Last Supper and how such a host may have known how to prepare for the Lord’s Passover?
Ultimately, we may never know. Some think it may have been Nicodemus, Joseph of Arimathea or the mother of John Mark.
Here is what Wikipedia has to say on the subject:
Location of the Last Supper
According to later tradition, the Last Supper took place in what is today called The Room of the Last Supper on Mount Zion, just outside the walls of the Old City of Jerusalem, and is traditionally known as The Upper Room. This is based on the account in the Synoptic Gospels that states that Jesus had instructed two disciples (Luke 22:8 specifies that Jesus sent Peter and John) to go to "the city" to meet "a man carrying a jar of water", who would lead them to a house, where they would find "a large upper room furnished and ready".[Mark 14:13–15] In this upper room they "prepare the Passover".
Bargil Pixner claims the original site is located beneath the current structure of the Cenacle on Mount Zion.
No more specific indication of the location is given in the New Testament, and the "city" referred to may be a suburb of Jerusalem, such as Bethany, rather than Jerusalem itself. The traditional location is in an area that, according to archaeology, had a large Essene community, a point made by scholars who suspect a link between Jesus and the group.
Saint Mark's Syrian Orthodox Church in Jerusalem is another possible site for the room in which the Last Supper was held, and contains a Christian stone inscription testifying to early reverence for that spot. Certainly the room they have is older than that of the current coenaculum (crusader – 12th century) and as the room is now underground the relative altitude is correct (the streets of 1st century Jerusalem were at least twelve feet (3.7 metres) lower than those of today, so any true building of that time would have even its upper story currently under the earth). They also have a revered Icon of the Virgin Mary, reputedly painted from life by St Luke.
Building upon this general information the Catholic Encyclopedia adds a few more insights into the question:
Place of the Last Supper
The owner of the house in which was the upper room of the Last Supper is not mentioned in Scripture; but he must have been one of the disciples, since Christ bids Peter and John say, "The Master says". Some say it was Nicodemus, or Joseph of Arimathea, or the mother of John Mark. The hall was large and furnished as a dining-room. In it Christ showed Himself after His Resurrection; here took place the election of Matthias to the Apostolate and the sending of the Holy Ghost; here the first Christians assembled for the breaking of bread; hither Peter and John came when they had given testimony after the cure of the man born lame, and Peter after his liberation from prison; here perhaps was the council of the Apostles held. It was for awhile the only church in Jerusalem, the mother of all churches, known as the Church of the Apostles or of Sion. It was visited in 404 by St. Paula of Rome. In the eleventh century it was destroyed by the Saracens, later rebuilt and given to the care of the Augustinians. Restored after a second destruction, it was placed in charge of the Franciscans, who were driven out in 1561. At present it is a Moslem mosque.
For those who place stock in the private revelations of Blessed Catherine Emmerich, she reveals that the room where the Last Supper was celebrated belonged to both Nicodemus and Joseph of Arimathea. As both were friends of Jesus it would be easily understood how they knew to prepare the Upper Room the way Jesus desired it to be.
I saw the two Apostles ascending towards Jerusalem, along a ravine, to the south of the Temple, and in the direction of the north side of Sion. On the southern side of the mountain on which the Temple stood, there were some rows of houses; and they walked opposite these houses, following the stream of an intervening torrent. When they had reached the summit of Mount Sion, which is higher than the mountain of the Temple, they turned their steps towards the south, and, just at the beginning of a small ascent, met the man who had been named to them; they followed and spoke to him as Jesus had commanded. He was much gratified by their words, and answered, that a supper had already been ordered to be prepared at his house (probably by Nicodemus), but that he had not been aware for whom, and was delighted to learn that it was for Jesus. This man's name was Heli, and he was the brother-in-law of Zachary of Hebron, in whose house Jesus had in the preceding year announced the death of John the Baptist. He had only one son, who was a Levite, and a friend of St. Luke, before the latter was called by our Lord, and five daughters, all of whom were unmarried. He went up every year with his servants for the festival of the Pasch, hired a room and prepared the Pasch for persons who had no friend in the town to lodge with. This year he had hired a supper-room which belonged to Nicodemus and Joseph of Arimathea. He showed the two Apostles its position and interior arrangement. - Preparations for the Pasch
But all this does not paint a complete picture. Scriptures also reveal that both Nicodemus and Joseph of Arimathea helped in Jesus’s burial in a rather hastily manner!
38 And after these things, Joseph of Arimathea (because he was a disciple of Jesus, but secretly for fear of the Jews) besought Pilate that he might take away the body of Jesus. And Pilate gave leave. He came therefore, and took the body of Jesus.
39 And Nicodemus also came, (he who at the first came to Jesus by night,) bringing a mixture of myrrh and aloes, about an hundred pound weight.
40 They took therefore the body of Jesus, and bound it in linen cloths, with the spices, as the manner of the Jews is to bury.
41 Now there was in the place where he was crucified, a garden; and in the garden a new sepulchre, wherein no man yet had been laid.
42 There, therefore, because of the parasceve of the Jews, they laid Jesus, because the sepulchre was nigh at hand.- John 19:38-42
There are some who put forth the theory that Shroud of Turin could have been the table cloth used in the Upper Room of the Last Supper!
The Shroud of Turin is not only the burial cloth of Jesus - it may also be the table cloth used at the Last Supper, according to evidence presented by John and Rebecca Jackson, who lead the Turin Shroud Center in Colorado. John Jackson, who was part of the 1978 team that investigated the shroud, said there are drips of liquid that could have been spilled on the cloth during the Passover meal shared by Jesus and the apostles. - Mysteries of the Shroud