When I think of being politically active as a stranger and alien, I frame it in terms of obeying God's direction to the people of Israel found in Jeremiah 29:7:
Also, seek the peace and prosperity of the city to which I have carried you into exile. Pray to the Lord for it, because if it prospers, you too will prosper.
Believers will differ on what it means "to seek the peace" of our city in terms of the appropriate level of engagement. Personally, I believe that being the citizen of a democratic country, I have the God-given power to influence it by my vote. Accordingly, I am responsible for how I exercise that power or even if I fail to exercise it when I should.
Beyond that, I'm not sure. However I became involved, I would want to
1) follow kingdom principles of truth, integrity, mercy and love,
2) be able to be open about my faith (without campaigning on it),
3) not to get sucked into the mindset of worldly wisdom described in James 3:14-16 (which comes so easily in an adversarial system),
4) not become so engaged that I lose a sense of detachment as a resident alien and
5) not confuse the goals or the party with which I am working for God's kingdom.
From the kingdom perspective, our party divisions are as arbitrary as the Blue and the Green factions in the Roman chariot races seem to us today. God's kingdom will come in full only when Jesus comes.
Christians in politics are not engaged in kingdom building so much as in palliative care for a lost society.
However, I believe that through political engagement believers can put feet to that part of the Lord's Prayer which says "Thy will be done on earth as it is in heaven" by working toward peace and righteousness in however limited form. Christians may not all agree on the ideal social state and we should cut each other some slack on that, but I think we can agree to work against gross injustices and abuses in the city of our exile.
Regarding point (3), I think that it takes a special grace for a believer to be involved in politics while keeping his balance spiritually. Just as it is a special challenge for a believer to be rich or to exercise earthly power in any form.