Yes, the database option is what you need to run apex applications. If you look at https://cloud.oracle.com, there is a place to sign up for a free 30 day trial but I haven't done it yet so I don't know if they are fulfilling these requests yet.
Did you get your account activated? I requested for a 30-day trial today and was hoping to see an email immediately with activation details! Can anyone share their experience on how long this would take?
Oracle Application Express (APEX) is distributed with all editions of the Oracle database since 2004 and database 10g. APEX is a no cost feature of the Oracle database. The easiest way to download apex is to visit http://apex.oracle.com and click download. This will link you to our Oracle Technology Network (OTN) download site. You can use APEX on our no cost Oracle XE database as well as our most expensive Oracle Enterprise Edition.