Nektarios,
It's my view that at the present time, any organized "apostolate" or mission to the non-Orthodox is crippled immensly by the scandals and heresies within the (at least nominally/traditionally) Orthodox world itself. It requires a tremendous revelation from God, for someone to find their way to the Church in our times - both because of the long term imposters to "Christianity" that have existed for centuries, but also because there is now the scandal of churches with the name "Orthodox" legally in their title that are in fact nothing of the sort.
I do not think until this begins to be sorted out (or at the very least genuine Orthodox can begin to get their act together and sort out their affairs, at least a little bit) that there will be much in the way of an organized, corporate effort to either missionize the west, and certainly nothing in terms of an organized attempt to convert people from particular heresies. Rather, for the time being, the trickle of genuine conversions will be just that - it will happen mostly on an individual basis, and with surprisingly little help from mortal hands.
That may not sound very optomistic, but I think this is the state of things.
Individually, the best thing we can do is try to work out our own salvation. There's a profound deceit in thinking you can save anyone's soul, when you'll be fortunate to leave this life with genuine love for God and the feeling of sadness for your sins in your heart. This is one of the fruits of deceit inherit in many of the so called papist "saints" - the delusion that their primary orientation was to go and "save souls" by their meritorious works...the deceit that they were "good people", even in a position to be doing such things. OTOH, the great Saints of the Church of Christ, if anything, only became more aware of their vileness in God's sight as they progressed on their way, the Lord before Whom even the highest Heavens are impure. Hence, why St.Seraphim of Sarov taught (in his famous saying) that if one acquires the Holy Spirit (through a life of repentence and dilegence toward the things of God) he will save many around him.
Seraphim