Problems with rapid melting ice cream. Making it without a machine I know this has been discussed before, but I am approaching the problem from another angle: I don't have an ice cream machine.

So I made Alton Browns Philedelphia style ice cream 

I made the base, put into a freezer bag, froze it for approximately 7 hours, removed it from the freezer, into a food processor for 30 seconds or until the texture was creamy and then back into the freezer where I kept it overnight.

The result was a very delicious ice cream. However, it melted very easily and I am wondering what I can do to prevent that. It is my understanding that fast melting is a result of too much overrun. But I didn't have a machine so the only overrun is the result of my food processor breaking up the crystals?

I might have stored the base for too long in the freezer. After 7 hours, a lot of crystallization has taken place and I could hear the blade in the food processor beating the crystals while the mixture was in there. For next time, I thought about storing the mixture in the food processor container and just put it into the food processor every 30-60 minutes while it is being frozen. 

What can I do to prevent the rapid melting of the Philadelphia style ice cream, given that I don't incorporate an ice cream machine in my process?
Most the time when ice cream is melting faster than it should, the problem is over-churning it. Like you said since that is eliminated due to the fact you don't have an ice cream machine. 

I would suggest a method I use: keep an old coffee tin laying around (one of the metal ones). Wash it out and dry. Then stick it in the freezer for 10-15 minutes or until it's cooled. Then make your batter directly in the can, then back to the freezer for a few hours. The coffee can helps insulate the batter and keep it cold even when it is out of the freezer. 

Then use the same food processor method as before. Scoop the ice cream mixture back into the coffee can and let it set in the freezer over night. The next day you will have perfectly scoopable ice cream that doesn't melt too fast.