Yang getting a new arm might seem like a good idea to cure her depression, but not really. It would be like putting band-aids on a open wound. A new arm won't bring Pyrrha back to life. It won't get rid of the Grimm infestation at Beacon academy and Vale. It won't make Blake come back to her friends. It won't set Weiss free from her father. Of course a new arm would help, but it's not what Yang needs the most right now. Yang was once an happy and active girl, but she has now been reduced to quiet one who wants to be left alone moping. Her once bright, inner fire has been reduced to just a few embers. She's just a shadow of her former self. It seems she has given up, and no longer have any interest in training or traveling. She's likely going to receive some pep talk from her father, and mother too. Both of them will use different methods, and both will be effective in their own way. But besides her parents, Yang needs more to become motivated again, and that is from her sister. If Yang finds out that her sister might be in danger, or that she's tracking down Cinder, then Yang would likely jump out of bed and rush out from Patch. Ace is right that telling her that she needs to get better won't work. She has to realize that herself. We have no idea what RT are planning, but we should assume that Yang will at some point get a robotic arm, but not necessarily in volume 4. Cybernetic limbs are probably very rare and expensive in Remnant. Not just anyone can get it. It's probably only Atlas that has the technology. I guess it's going to be Weiss who will help get the arm, using her family's connections and resources. Yang is a brawler. She needs both her fists, even if one of them is robotic.