What would a Trump foreign policy be like? Given his background in business, we can look there for some clues.
It is common for businesses to have relations with their competitors. For example, while Apple and Samsung are indeed long-term competitors, Apple purchases screens for its iPhones from Samsung. The two companies do business together as long as it is in their interests to do so, all the while being competitors.
Now apply this to Syria. Under Putin and Obama, Russia and the US are essentially fighting a proxy war: Russia is backing Bashar al-Assad while we are supporting the opposition rebels. If Trump should side with Russia, then the rebels would lose most certainly their main supplier of funds and arms and other support.
Like Apple and Samsung, we would be allies with Russia, but only on this one issue.
Once the rebels are gone, this temporary alliance ("one issue alliance") between Russia and the US will cease.
This is both similar and different from the relationship we had with Soviet Russia during World War II: similar in that two ideological enemies team up to fight a common enemy; different in that we need not paint Russia in a positive light, like was done during WW2 by Hollywood.
This "Trumpian" foreign policy, if Trump indeed uses this approach, will cause deep consternation with NATO countries and other traditional allies. Those alliances are founded on the idea that Russia can do no good, while the allies can do only good. Convincing them that these one issue alliances are indeed temporary will be the second most difficult part to implement.
The most difficult part? Preventing this from dissolving into mercantilism.