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The Code of Canons of the Eastern Churches codifies common portions of the canon law of all the Eastern Catholic Churches within the (Roman) Catholic Church.
Canon 883 says
The Christian faithful who are outside the territorial boundaries of their own Church sui juris can adopt fully for themselves the feast days and days of penance which are in force where they are staying.
This shows that someone whose own Church is an Eastern Rite Church, but who lives in a Western country, is free to observe Lent according to the Latin rite instead of the rite of his own Church.
The question is about someone attending a Latin rite Church, rather than living in Latin rite territory, but in practice most Eastern Rite Catholics regularly attending a Latin Rite parish will do so because of where they live, and so I offer this answer.
In years when Easter is the same East and West, Clean Monday is 2 days prior to Western Ash Wednesday and Eastern Lent ends 2 days before Palm Sunday. This year Eastern Easter is one eeek later. There are obvious disadvantages in fasting according to one calendar but using the liturgy of another as one could be fasting in preparation for Easter even while celebrating it in church.