What about Red Hat's OpenShift? It's not quite shared hosting but it should fit your needs.
OpenShift is Red Hat's free, auto-scaling Platform as a Service (PaaS)
for applications. As an application platform in the cloud, OpenShift
manages the stack so you can focus on your code.
No need to bother with component installation. Just pick the Java cartridge and then after that, add the Postgresql cartridge ( that already includes the PostGIS plugin ). You can check the list of supported features here.
Oh and did I tell you that you could try it for free? They have a free tier which supports up to three gears.
A gear is a resource constrained container that runs one or more
user-specified software stacks, also known as cartridges. Each gear
has a limited amount of RAM and disk space. If an application needs
more resources, it can use multiple gears.
Since your components are basically Java, PostGIS and Tomcat, that should fit right in.
I hope that works for you :)