I recommend moving to another city.
In case that just isn’t convenient, there are a handful of things you could do, but none of them come close to actually immersing yourself in the Liturgy.
(1) Buy a chotki, if you can find an Orthodox bookstore nearby. Or even if not, just set a given amount of time each day to pray the Jesus Prayer.
(2) I like the prayerbook by Fr. Nomikos Vaporis, which you can get through inter-library loan. Also, find a copy of the Akathist. (Before Vatican II, the Akathist had 25 days indulgence from the Pope, so plenty of motivation for Roman Catholics to pray it anyway - and one can always pray for the graces of an indulgence even after the Church got stingy with them.)
(3) Use “The Way of a Pilgrim”, and then the Philokalia for your spiritual reading.
(4) Buy some icons - just so long as you avoid the very flourishing company that calls itself “Monastery Icons”. They appear to be Catholic, but the monks that write the icons belong to a group known as the Gnostic Orthodox Church.
(5) Fast. No meat, eggs, fish, or dairy on Wednesdays and Fridays, and during fast periods (Great Lent, three weeks before the feast of Sts. Peter and Paul, two weeks before the Assumption/Dormition, and between the feast of St. Philip and Christmas).
(6) One way to immerse yourself in the Eastern atmosphere, though it should certainly not be taken as a substitute for prayer, is by listening to the sacred music (liturgical and otherwise) of the East. There are lots of CDs of choral liturgies, which the Russians in particular have a very strong tradition for. Rachmaninov, Tchaikovsky, and Ippolitov-Ivanov all composed both a Vespers and a Divine Liturgy; Alexander Gretchaninov composed at least 4 Liturgies and a Vespers, Pavel Chesnokov’s Requiem No. 2 will give you goosebumps. Alexander Glazunov also composed a very beautiful non-liturgical work called “The King of the Jews”. Sergei Khvoshchinsky composed a Bogoroditse Devo which is worth dying to listen to (recorded by the Rose Ensemble, in their CD “Fire of the Soul”). Stefan Mokranjac composed a Liturgy and at least 2 Requiems; Arvo Part (who is Orthodox) composes mostly music for the Roman Rite but with a deeply Eastern spirit. Sir John Tavener is also Orthodox in some fashion but draws much of his inspiration from (non-Christian) Eastern religions; his Akathist of Thanksgiving is good though, and his “Veil of the Temple” is very moving. Alfred Schnittke (who I believe was a Catholic of the Roman Rite despite being Russian) has composed some non-liturgical sacred music in Russian, including Russian prayers like the Bogoroditse Devo).
There are also CDs of the Divine Liturgy celebrated by the Old Believers of Gervais, Oregon, and lots of recordings you can get on the Russian Orthodox webpage and on YouTube. Anything on YouTube posted by vagos2006 is going to be very, very good.
(7) Meet a Byzantine Catholic. Then you’ll know someone who pretty much knows or is one or two steps removed from pretty much every other one.
We tend to be a very hospitable and tight-knit community.
(8) The only other advice I could give would be to study the theology of the Eastern Church - immerse yourself in St. Symeon the New Theologian and St. Gregory and St. Maximos the Confessor. And even that is still going to fall way short of actually seeing a Divine Liturgy.
