I Would vote no.
Building a robotic arm yourself is quite difficult, but like others said, is possible. However, this will mean that you spend a lot of time building this and not focussing on your goal of learning about neural networks.
If you want to start with neural networks, I would vote to start with image recognition, or basic numerical problem solving, all of which can be done in software, without the need for mechanics.
If you want to test a neural network on a robot, I would vote to take an existing robot platform (can be an arm, or a rolling platform, or anything else) and load your neural network on the controller.
I think this way, you will learn much more in less time. But if you want to learn how to build an arm, by all means, go ahead!