I am not aware of American pricing/products available, and can not really recommend as shopping questions are not allowed here, but I will give some generic advice.
Cheap NAS units are usually not very good and have very slow transfer speeds - that being said, unless your friend has a cable/fibre connection, even cheap NAS units will most likely be faster than the speed of the connection.
All this being said, typically the cheaper NAS units are very limited on protocols. It will most likely support SMB/Samba, but, opening these ports up across the internet is asking for trouble and is a very bad idea.
I think without a VPN/similar, you are going to be really stuck for "mountable".
You can map and do a bit with FTP, but standard FTP security is debatable. As long as you trust your friend (after all, you are leaving your files there - unless you are also wanting to encrypt the NAS).... this should work fine.
If you do not like standard FTP, I am not sure Windows natively supports SFTP and if it does, a low end NAS unit is unlikely to support it.
All this being said, on a low budget, quite frankly, the best thing you can probably do is use an old computer or even better, buy an atom based board and a drive and load on Windows Server if you have a license (spare or through a program such as dreamspark e.t.c), then set up VPN (follow my guide). Then, you can just natively use Windows File Sharing - initial setup simplicity is debatable, but, this will be the most secure and best solution in my books.
I know you said no to VPN, but, I assume this is because you do not want to make changes to your friend's network or buy an additional box. If however you are willing to forward some ports, what difference does it make if you instead of the sharing ports, you forward the VPN ports to this box. The VPN ports will be all you need to do.
If you can find a NAS unit that has both Samba and VPN, then, you can cut off Windows Server all together, but, I do not know of such a device. Similarly, you can most likely use Linux instead of Windows Server in the above example, but, I don't know where to get started there.
Hope this helps your!