Question on hanging new door in concrete block wall
This is a standard application for a pre-hung door, so you're on the right track. Building a custom jamb would get you the perfect size for your opening, but unless you have a router and a hinge template for mortising the jamb, that project is kind of a pain. So if time is money, go with the pre-hung.
Widening the doorway would require renting a masonry saw and it's a loud and dusty operation. Cutting the block may also compromise the structure, so I would stick with reducing rather than expanding your rough opening.
If you haven't done this already, pull off the trim from the around the door and you should be able to measure your rough opening exactly, and purchase the biggest door that will fit.
If it is indeed 39 1/4, that's not enough for a 38" door, so you'll have to stick with the 36" door.
So now all you need to do is reduce your opening from 39 1/4 to approximately 38 inches, which as you mentioned would be 1&1/4 inches.
You can do 5/8 material on both sides, or do 3/4 on the hinge side and 1/2 on the knob side.
You'll need a hammer drill and masonry screws to attach the rough lumber.
Another thing I would suggest is that once you have the door in place, pull off the wood piece of door stop from the hinge side of the jamb. In that space, put some masonry screws through the jamb, through the rough lumber and all the way into the cement block. Counter-sink these screws and cover them with the door stop. This will give your door some serious structure to hang off of, and will prevent any racking.