It sounds like you are looking for a minimalist approach to getting your app out there. That being the case, then there are a few things to consider.
Configurations:
It is definitely preferential to find a server that matches your development environment as closely as possible to limit the amount of retesting that you have to do. Changes in the version of tomcat or the DB will not likely result in changes to your code, but can result in you spending a lot of setup time. Try to match things as closely as possible.
Security:
I always worry about security. Maybe I am paranoid, but I live by a general rule here: if it is on the internet, someone will try to hack it. You should definitely layer security into your application as run it on https for any portions that are sensitive or require login data.
Hosting:
There are many companies that host tomcat/mysql servers. It really depends on finding a price that you like and what added services the host offers. Do they backup the system? Do they apply security patches to the OS? Think of it as a sliding scale: the more you want them to do, the greater the cost. The more you do, the lower the cost. It sounds like you want a provider that maintains the OS and general setup for you. your access to space on the box might be limited, be sure that you get access to the tomcat directory, you might need it.