I've written an easy web application mostly filled with static webpages with several you-tube embeddings. Couple of forms and so forth. All of this is developed using CodeIgniter.

The website is much like Academic Earth. With videos offered from Youtube. It's got 400 customers only 30-40 are active.

I'm not thinking about hosting that is shared and wish a piece of equipment with root access. My budget enables me to employ RackSpace cloud server with 256MB Ram.

Could it be sufficient to operate my application?

Or if this should question continue ServerFault.

I have had positive results running applications on small VPS hosts like Slicehost and Linode.

You'll most likely be fine using the apache/php/mysql out of your distribution's package manager. I have had great results producing my very own Amplifier stack from source and fine-tuning things. With PHP 5.3.3, PHP-FPM (FastCGI Process Manager) is built-in, and offers far better FastCGI support.

Running PHP under FastCGI could be adventageous in low-memory situations, as you're able to then run apache using the worker MPM, which is a lot more efficient, memory-smart than running prefork-apache/mod_php

Observe that presently, Linode provides you with a 512MB server for $20/month. However, my experience continues to be that Linode has semi-frequent network issues (see http://status.linode.com), but otherwise they have been great.

If all you are able afford is $20/month, you most likely can't do far better. Another nice factor about these VPS hosts is the fact that if you want to grow later, it just takes a few clicks inside a user interface and you will upsize your server.