You'd need to go through Syria to get to Turkey anyway. Lebanon borders only Syria and Israel, with the Mediterranean Sea to the west.
Short of hunkering down, I think the best bet would be to try to get to Amman via Damascus, if you can't be guaranteed a flight out of Damascus. I wouldn't want to get stranded in Damascus.
"I think the best bet would be to try to get to Amman via Damascus,"
Now is not the time to be thinking about moving anywhere in Beruit.
Let alone to Amman.