You can, of course, run your own PostgreSQL instance. There are hosting providers listed on the PostgreSQL Web site, or you could do something like rent a Linode and install PostgreSQL on that (hey, they have really reasonable prices—I use them for all sorts of VPS systems).
If you have access to the IPv6 address space from your Android device and you know of anyone who runs virtual machines for a hobby and can give you a VM with an IPv6 address to play with for free, that might be another option.
The only free or extremely low-cost places that I can find that support PostgreSQL databases do not permit you to do so from a public address; rather they require that you access it from the Web hosting that they also provide for free or very low-cost.
EDIT: What I didn't make explicit at first was the idea that you could just run it on your home network and use Android to connect to it. That is what I would do, personally, because then you have complete control of the whole process, and when you are developing things, that's the best way to do it IMHO.