Should I reinforce the cupboard back before mounting it to the wall?
It depends on whether the shelves are attached to the back of the cupboard or the sides. They are effectively transferring weight from the contents to their attachments, which is why most cupboards with shelves attached to sides have the sides attached to the walls using some form of bracket.

One way to ensure that the cupboard is supported properly is to attach a piece of lumber to the wall along the bottom of the cupboards, they then effective sit on it. This of course assumes you have the space to do this, and that the lumber will not look bad if viewing it from the bottom.

You can do the same thing towards the top of the back by having a piece of lumber that screws into the sides (from the back) and runs the length of the cupboard. This can then be screwed through from inside the cupboard to provide extra strength. The downside of this is that it will mean the cupboard will sit away from the wall the thickness of the lumber. You would also put this piece towards the top of the back of the cupboard, but usually not right at the top, although having at the top would not be a problem. There needs to be enough space to screw through.

Combined, the lumber below the cupboard supports the weight of the cupboard and contents, with the piece at the top preventing it from falling forward.