maybe. :-)
it will depend on which transformers you've used in your workspace. if you've used anything that's been updated for FME 2013 then it won't work.
having said that, it's possible ( but not recommended ) to manually edit a workspace in a text editor to specify which version of a particular transformer should be used if you're having trouble with just one or two.
You'll have to be pretty careful here, so I'd suggest that you install FME Desktop 2012 on a local machine somewhere and open your 2013 workspace in Desktop 2012. You'll find out pretty quickly if there's a problem as any newer Transformer versions that have been used will be removed from the workspace. See here for more information about transformer versions:
http://fmepedia.safe.com/articles/How_To/Transformer-Versions
http://fmepedia.safe.com/articles/How_To/Transformer-Versioning
There is another possible solution that is a half way house and though I've never done this between major versions you may want to look into just upgrading your FME Server 'engine' as it's possible to do that without completly redeploying Server. You just use the usual FME 2013 installer and choose to install just the engine, rather than the whole of FME Server. Once done you'll see on the admin interface what the engine build number is reported as. It should be pretty quick to do and even roll back if you need to without tampering with the rest of Server. More details here, and it's reassuring to see this FMEPedia page has also recently been updated, suggesting the info is pretty current:
http://fmepedia.safe.com/articles/How_To/Upgrading-FME-Engines
Best of luck, Dave
As of 2012, I believe FME stores the "version" of the workspace that it was created it. It will warn you if you try and use it with an earlier version (but you can anyway).
Also:
It'll run fine as long as you don't have any 2013 version transformers in it. If you do, FME will tell you when you open it. See Daves response links for that.