Are there any medieval (from the 5th to the 15th century) directives (some kind of manual or rubrics) about what to do/pray/think during Eucharist Adoration?
Eucharistic Adoration as it is known today didn't exist between the 5th and 15th centuries. Even Quantore (or forty hours devotions) did not begin until the 16th century. Apart from religious houses, perpetual Eucharistic adoration did not begin in earnest with the laity until the 18th and 19th centuries in Italy and France. (Wikipedia)
Are there any Catholic medieval directives (rubrics) about what to do/pray/think during eucharistic adoration?
There are certainly some directives about what to do during Eucharist Adoration, but they are extremely hard to find , if not quasi impossible to find.
(This is going to be an incomplete answer, at best since genuine original source are lacking in this subject matter.)
Eucharistic Adoration does not originate in the Middle Ages, not with Corpus Christi nor with Lateran IV. It has a lengthy pedigree that takes us from the Middle Ages back through the Eucharistic controversies of the 11th century to the flowering of the monastic age and the desert fathers - right on down to the New Testament itself. Those who consider Eucharistic Adoration a strange aberration of the Middle Ages should consider that Eucharistic Adoration not only was practiced in the early Church, but that the principles upon which Eucharistic Adoration is based are all developed from apostolic teaching.
First of all, let us deal with when Eucharist Adoration was first historically documented.
Eucharistic adoration began in some form early in Church history but it was clearly promoted by a succession of popes beginning in the Middle Ages, including Pope Urban IV who established the Feast of Corpus Christi in the 13th century and Pope Pope Clement VIII who in 1592 instituted 40 Hours Devotion.
On 11 September 1226, in compliance with the wish of Louis VII, who had just been victorious over the Albigensians, the Blessed Sacrament, veiled, was exposed in the Chapel of the Holy Cross, as an act of thanksgiving. So great was the throng of adorers that the bishop, Pierre de Corbie, judged it expedient to continue the adoration by night, as well as by day, a proposal that was subsquently ratified by the approval of the Holy See.
Exposition and consequently adoration became comparatively general only in the fifteenth century. It is curious to note that these adorations were usually for some special reason, e.g. for the cure of a sick person, or, on the eve of an execution, in the hope that the condemned would die a happy death. The Order of the "Religiosi bianchi del corpo di Gesù Christo," a Benedictine reform, united to Cîteaux in 1393, and approved later as a separate community, devoted themselves to the adoration of the Blessed Sacrament. Philip II of Spain founded in the Escorial the Vigil of the Blessed Sacrament, religious in successive pairs remaining constantly, night and day, before the Blessed Sacrament. But, practically, the devotion of the Forty Hours, begun in 1534, and officially established in 1592, developed the really general Perpetual Adoration, spreading as it did from the Adoration in one or more churches in Rome until it gradually extended throughout the world, so that it may be truly said that during every hour of the year the Blessed Sacrament, solemnly exposed is adored by multitudes of the faithful. In 1641 Baron de Renty, famous for devotion to the Blessed Sacrament, founded in St. Paul's parish, in Paris an association of ladies for practically a Perpetual Adoration; and, in 1648, at St-Sulpice the Perpetual Adoration, day and night, was established as a reparation for an outrage committed by thieves against the Sacred Host. - Perpetual Adoration (Catholic Encyclopedia)
From this point on Eucharist Devotion increased in various locations amongst the faithful in various countries.
The lay practice of adoration formally began in Avignon, France on September 11, 1226. To celebrate and give thanks for the victory over the Albigensians, King Louis VII of France asked the Bishop of Avignon to have the Blessed Sacrament exposed in the Chapel of the Holy Cross. The throng of adorers was so great that the bishop decided to have the adoration continue day and night.
During the Middle Ages, many more of the faithful began to adore the Blessed Sacrament apart from the Mass. At first, the custom was to worship the host reserved in the tabernacle. Eventually, some came to practice the devotion with the tabernacle doors open. Later still, solemn exposition of the host, in a monstrance, became the norm.
The practice spread through Europe and culminated in the establishment of the Feast of Corpus Christi—Latin for “the Body of Christ”—in 1264. The feast itself, now celebrated each June, helped spread the devotion. - History of Eucharist Adoration
Wikipedia has this to add at this moment:
In 1264, Pope Urban IV instituted the feast of Corpus Christi ("the Body of Christ") with the publication of the papal bull Transiturus. He asked the Dominican theologian Thomas Aquinas to write the texts for the Mass and Office of the feast. This included such famous hymn as Panis angelicus, and Verbum Supernum Prodiens the last two strophes of which form the Benediction hymn O Salutaris Hostia. The last two verses of Pange Lingua are sung as the hymn Tantum Ergo, also used at Benediction.
As of the fourteenth century in the Western Church, devotions began to focus on the Eucharistic gifts as the objective presence of the risen Christ and the Host began to be elevated during the liturgy for the purpose of adoration, as well as to be seen by the congregation since the priest stood facing the same direction in front of the altar.
A common early practice of adoration known as Quarantore (literary forty hours) started in the 16th century. It is an exercise of devotion in which continuous prayer is made for forty hours before the exposed Blessed Sacrament. This practice started in Milan in the 1530s and 1540s by Capuchins such as Giuseppe da Fermo who promoted long periods of adoration. From Northern Italy it was carried to elsewhere in Europe by the Capuchins and Jesuit.
Now let us start with medieval directives for Eucharist Adoration which I Will try to complete as time permits:
In the Middle Ages, the worship of the Eucharist was ultimately an expression of reverence toward the holiness of God. Following consecration, the maiestas Domini, divine majesty became present on the altar. According to Jungmann, the elevation following the consecration came in use in Paris, in the 12th century, allowing the congregation to worship Christ in the Eucharist. Starting from the 13th century, we have solid data to prove that elevation followed the words of consecration. However, Nußbaum shows that in Germany elevation was practiced already before the 12th century. The first synodal resolution on the practice of elevatio following consecration was issued by the Council of Paris, in the time of Bishop Eudes de Sully (1196–1208). In 1219, Pope Honorius III commends the practice of elevation in his letter to the English bishops. In the 13th century, Saint Albert the Great already regards elevation a common practice. According to Browe, the two additional presentations of the consecrated elements following the great elevation – during the doxology following the canon and that following the Agnus Dei – originate in this practice.
Starting with the 13th century, during elevation, the congregation made the sign of the cross or beat their breast as a sign of reverence before Christ. Later, following elevation silent prayers and hymns were prayed. Ringing hand-held bells and the church bell to call the attention of the congregation was not yet a general custom in this period. As the faithful desired to see the elevated Eucharist, candles had to be lighted at early morning masses for the Eucharistic species to be visible. In the 14th century Carmelite monasteries the elevated host was also venerated with incense. In the 16th century Spanish and English churches, a black veil was hung in front of the altar to enhance the visibility of the white host against the dark background. This practice became widespread in France as well; in the churches of Chartres, Rouen, and Orléans purple veils were used as background. The priest celebrating the mass moved the elevated host left and right to ensure its visibility.
Elevating the chalice was a later and slower development. The first clues lead to France. The practice is mentioned in Wilhelm Duranti[s]’s (Guillaume Durand, +1296) Rationale divinorum officiorum. According to the missals, elevating the chalice became widespread starting from the 15th century. The Council of Constance presents the elevation of the chalice as a self-evident practice. The 1570 Roman Missal of Pope Pius V already mentions double elevatio. The diversity of the forms of worship glorifies Christ present in the Eucharist and the faithful grasped every opportunity to see and worship the sacrament. - Eucharistic Devotion
So are there any Catholic medieval directives (rubrics) about what to do/pray/think during Eucharist Adoration?
The usual position for the faithful to pray while exposition of the Blessed Sacrament is kneeling in prayer on both knees. I grew up with this custom in full force. Nowadays, many simple sit down while the Blessed Sacrament is exposed. All genuflect (bending both knees) when adoring the Blessed Sacrament unveiled, as at Expositions. This liturgical custom dates back to the Medieval Catholic Church.
Historically, the Church has never told the faithful what order how to pray while in church, whether the Blessed Sacrament is exposed or not.
Since illiteracy was quite high during the Middle Ages, it is logical that many of the faithful would pray the rosary. Those who could read possibly would read some biblical passages.
Certainly, some religious, such as Benedictine monks would pray the Divine Office while the Eucharist Adoration is going on. This is an actual monastic tradition in some regions.