I'm going to be a little wordy here, but I think it's justified : )
To celebrate the entry Byzantine Rite divine office requires about twenty LARGE books
] of hymns and prayers. The closest we have in modern English are the books put out by the Melkites, but these often differ in quite a few ways from the Slavic liturgical tradition used by Ukrainians and Subcarpathian Ruthenians.
This was a problem in small churches even in Europe, and one solution was to use the small Byzantine collection called the General Menaion: a set of generic hymns for each class of saint, when do didn't have access to a full twelve-volume Menaion with hymns for the liturgical year.
The new (2006) Divine Liturgies book in the Byzantine Catholic Church contained such a general menaion for the Divine Liturgy, so that if prophet is celebrated on a particular day, one can sing the troparion, kontakion, prokeimenon, and alleluia for a prophet, inserting only the name where appropriate.
NOW, the MCI Menaion mentioned by the original poster was intended as a supplement to the Divine Liturgies book. According the MCI website publications page, it contains
"... the synaxarion (names of the saints and events commorated on each day, with brief explanations), together with the troparia, kontakia, prokeimena, and communion hymns assigned to each day in the calendar."
This is basic information needed for the celebration of the Divine Liturgy for the SPECIFIC saint of the day, including prefestive and postfestive hymns, WITH MUSIC, and a little information about the liturgical commemoration of the day. The troparia and kontakia are mostly taken from the corresponding section of the Book of Matins of the Sisters of Saint Basil the Great.
To celebrate Vespers and Matins, you also need the additional hymns called stichera
]. These ARE available for the Byzantine Catholic Church, though not yet in a complete set.
For example, at Sunday Vespers (on Saturday evenings), the MCI Sunday Vespers book includes the complete set of Sunday stichera at the Lamplighting Psalms and the Aposticha, in all eight tones, with music. On an ordinary Sunday, these would be supplemented with a small number of hymns for the saint of the day. The MCI website has a leaflet for every Saturday of the year with these hymns. Similar leaflets are available for each major feast.
For ordinary (weekday) Vespers, the MCI Daily Vespers book contains the weekday stichera in all eight tones for the Lamplighting Psalms and Aposticha, together with a general menaion for classes of saints. So one might combine, for example, the stichera for Thursday in Tone 5 with the hymns for a single apostle. The preface explains how this is done.
(The MCI Sunday Matins book contains all the hymns needed for Matins on an Ordinary Sunday, supplemented by leaflets for major feasts and the Sundays of the Great Fast. A book for daily Matins is in preparation, and I am working on the common of saints for that now, based on models from the Eparchy of Mukachevo.)
For further details, Father David Petras' Annual Typikon and Common Typikon (available from the Byzantine Seminary Press) are invaluable, and give ALL the details - but also assume that one has all the needed books.
By the way - the Byzantine Daily Office from Eastern Christian Publications is a wonderful resource, but:
1. It is NOT the complete Byzantine Divine Office; it omits the key services of Vespers and Matins, and contains instead the lesser Hours.
2. It does NOT provide the actual hymns (troparia and kontakia) for each day of the year from the Menaion (except for major feasts). Instead, it uses the common troparia and kontakia for classes of saints from the 2007 Divine Liturgies book. Thus, it sticks to "very official" texts but at the cost of a fair amount of repetition. It DOES include the synaxarion entry for each saint (from the MCI menaion).
Over the last eight years, in the course of providing music for Saturday evening Vespers, the MCI has assembled and set to music ALMOST all the Lamplighting stichera of the full Menaion, and later this year we plan to publish these as monthly collections, beginning in September, in editions with and without music.
P.S. I'm happy to discuss specific questions via e-mail at Plainchanter@gmail.com.