I need to deploy my web services and install Tomcat.
Which one is the best way?
To install OpenStack on my 32-bit machine as this project is just for test, or to use OpenShift?
But is it possible to use Tomcat with OpenShift?
Yes, it's possible to use Tomcat with OpenShift, through the use of JBoss EWS cartridges, explained in detail here.
First it's important to difference between OpenStack and OpenShift:
OpenStack provides “Infrastructure-as-a-Service”, or “IaaS”. It provides bootable virtual machines, networking, block storage, object storage, and so forth.
The OpenShift hosted service provides “Platform-as-a-Service” or “PaaS”. It provides the necessary parts to quickly deploy and run a LAMP application: the web server, application server, application runtimes and libraries, database service, and so forth.
You could also install and deploy your own PaaS services, using OpenShift Origin:
OpenShift Origin is the open source project of the software that enables the OpenShift hosted service. Using OpenShift Origin, you can build your own PaaS.
Complete note here.
So, if you need to test your services with Tomcat, I would go on with OpenShift, remember that you have 3 gears with your free account and you could easily and quickly perform the deploy of your application. Getting started online guide and free ebook.