Since it hasn't been mentioned yet, I've used openshift (RHEL 6) for a simple Perl based web-service. You can add cartridges for Perl (5.10. then), cronjobs, databases, etc. Registration and use of small resources is free (they have full access to your application of course).

If you need a modern Perl or i.e. Mojolicious, you can create your own DIY cardridges. I.e. OpenShift-DIY-Perl.

I am not sure, if arbitrary ports can be used (easily)?

You have ssh access to your server instance and can push new apps via git/ssh. Keep your application used frequently (once within 48hrs IIRC) though, otherwise it will be paused (hibernated?) and will need some seconds to restart, once you access it again.

Update (26-AUG-2017): Above refers to version 2 which will be discontinued after September 2017. Version 3 seems to support Perl 5.16 and newer as well as Dancer applications out of the box.