I want to be able to carry things in my arms for a longer period, carrying like carrying someone on your lower arms, half bicep curl position.
How do I increase endurance in this type of carrying?
The ideal exercises for training to carry stuff are, surprisingly, carrying stuff. On the assumption you can't find a willing person to be carried around, then something like sandbag work would be ideal. You can do sandbag carries for distance or time, in a bearhug or cradle position, practice picking it up off the floor, you can even try pressing it overhead for a bit of extra fun (note: I use fun in the sadistic sense). You can also use a keg, filled with sand or water, but a sandbag tends to be a little more forgiving.
You can use barbells and dumbbells to work some of the related musculature, but honestly the best way to get better at carrying something is to practice carrying something.
When you're carrying something (or someone), particularly in the cradle position, it's not just the arms you need to take into account, there's also a lot of stress on your posterior chain, spinal erectors and upper back, as well as your legs when you're actually in motion with the load, not to mention your core fires like crazy. If anything, I'd say that the arms are actually the least important element when it comes to carrying something in that position, as you can normally shift the weight around to move it more towards your own body, making it more mechanically advantageous to carry (lift it higher, lean back so it's more over your own center of gravity).
Besides sandbag work (seriously, if you've never tried picking up and carrying even something like a 75kg sandbag or keg, you're missing out), I'd look at a routine consisting of deadlifts, squats, rows, farmers walks, and if you have access to them, yoke carries and stone lifting. Yes, throw in some curls as well, because having stronger arms will help, then you may as well throw in some overhead presses to round things out (if this is starting to look like a strongman training program, there's a reason for that... a lot of their training revolves around lifting heaving things off the ground and carrying them).
But seriously, get out there and carry stuff, as Dan John says, it tends to be the missing element in most people's training programs.