An Old Rite Russian Greek-Catholic is a Greek-Catholic (which I trust I needn't define) who follows the Byzantine Liturgy in the form used in the Russian Church prior to the recasting (on pretext of translating and correcting - now why does that sound familiar?) - by the lamentable Patriarch Nicon of Moscow in the 1650s. "Old Rite" in this context refers to the pre-Niconian Russian usage of the Byzantine liturgical tradition; it lives on among the Russian Orthodox Old-Ritualists, a small number of Russian Greek-Catholics, and a few other enthusiasts here and there. Interest is growing, partly because with the collapse of the USSR it has become possible to publish the service-books of this tradition and some other studies and literature, partly because of a book-length study ( Russia, Ritual and Reform, Paul Meyendorff, Saint Vladimir's Seminary Press) and partly because the Russian Orthodox Old Rite parish in Erie, Pennsylvania is quite active and publishes significant materials in English - this is a "first" in the history of the Old Rite.
Hope this information is of help.
Incognitus