(Note: this is just based on a couple of hours desktop investigation, and may not fully correct. Others with more expertise in this area will know more !)
The original plan for Horizon 2020 had a budget of around 80bn Euros, with a focus on three strategic areas – Excellent Science, Competitive Industry and Tackling Societal Challenges. The Prezi presentation gives a good overview of the original intentions. In the outline of the programme written by the European Commission (pdf file published on November 2011), there was little or no mention of funding that would be for cultural heritage or humanities, whether digital or not
A number of amendments (thousands in fact) were suggested by the European Parliament (some, or maybe all, of them are available via this webpage). This included changes to incorporate humanities and cultural heritage, as a page on the Collections Trust webpage demonstrates. It’s not clear to me how many of these amendments were accepted, or whether the final text has still to be agreed on.
A revised budget of around 70bn has recently been agreed (June 26 2013) but still needs to be ratified by the European Parliament and European Council. This will be done in context of the overall EU budget decisions for the year 2014-2020 (called the Multi-Annual Financial Framework, MFF). The program is still scheduled to start on 1st January 2014, but this might be pushing as the European Commission reckons it requires six months to get a programme of funding up and running following ratification. Tweets have suggested there is specific funding for humanities and social sciences but I’ve not seen this substantiated elsewhere.
Horizon 2020 is only one of several funding Europe-wide programmes that the EU runs. This helpful list (scroll down the page for details) provides more details of the programmes. This includes the Connecting Europe Facility, which is the programme that was originally earmarked for funding Europeana and its related ecosystem of digital library projects and services.