CONGRATULATIONS!!!! On your new baby!!!
He might be sleeping too much during the day.
My second son couldn't sleep for more than 2 hours a night because we found out later that his ear tubes were NOT fully developed (he was born 4 weeks early and had pneumonia and stopped breathing), once we had tubes put in his ears - he slept the night through was a TOTALLY changed baby!!!
I could NOT do the CIO method - I just can't do that. I have let kids cry for a few minutes - but not longer than 5. Babies have NO concept of time.
Babies cry because they need SOMETHING - express your milk - give it to him in a bottle to make sure he's getting enough. It's TOTALLY possible that since you are sleep deprived, you are not making enough heavy milk and he's totally hungry.
At 5 months - he SHOULD be able to sleep at least 4 to 6 hours without waking. I would try expressing/pumping the milk to see how much he gets and go from there.
In the mean time - get sleep WHENEVER you can!!