I would definitely use Passenger for this. It's easy to set up and run, and performs well. Passenger takes care of spawning and killing child Rails processes to handle incoming requests.
The passenger module supplied by Ubuntu is ok, but slightly behind. A more recent package is available by installing from the semi-official repository. According to Passenger's install page:
Add the following line to the Third Party Software Sources:
deb http://apt.brightbox.net lucid main
(The simplest way to do that is to create a file in /etc/apt/sources.list.d/ containing the deb instruction, and then run 'apt-get update').
Once you've done this then you can install Phusion Passenger by running:
apt-get install libapache2-mod-passenger
I think the installer automatically enables the module and restarts Apache, but if not you should do that yourself.
Check the manual on how to deploy a Rails app, but you'll find it remarkably easy:
Allow from all
I would also advise you to look at running Ruby Enterprise Edition (ree) which is a modified Ruby interpreter, built by the same team as Passenger, that features reduced memory usage by about 33%. It's well tested and 100% compatible with Ruby 1.8.7. You'll find many people running it on production servers.