OpenShift(https://www.openshift.com/) may be the service you're looking for. I haven't tried it, but it looks nice.
I've only used it to create a small Python/Flask app http://wwos-bitmason.rhcloud.com/ but there are lots of examples and information on the OpenShift blog.
[Disclaimer: I work for Red Hat.]
It reuses Heroku's code for several buildpacks, particularly Ruby.
I know that it works really well because (disclaimer!) I worked on the Cloud Foundry Buildpacks team for Pivotal.
If you have a 12-factor app that pulls config from the environment, it should ideally not be too traumatic. Services are meant to be injected by the platform.
Apps should be the easiest part. Of the buildpacks: Ruby, Python and Node are based on Heroku's and are very nearly identical. Go is based on a community buildpack, PHP is a complete ground-up rewrite and Java is a ground-up rewrite maintained by the Spring team.
If you signup for PWS and you find things broken or missing, email me: jchester@pivotal.io, and I will find who can help you.
However you can work around it by hitting your server with an http request every 24 hours, or upgrade to the Bronze Plan which apparently has $0 base price anyway so I'm not sure what the downside is.
Also, for a relative novice, how easy is it to get a Rails or Sinatra app up and running?
My concern longer-term is that Openshift would go the same route as Heroku. Is there anything about their business model that might indicate that's not a major risk?