My phone system is Android 4.2.2, CM10.1. I have installed the GAPPS and Google+, but I still cannot sync photos from Picasa. Some people says syncing from Picasa is removed in 4.2+...really?
With the big Google Plus push this past year, Google have been slowly moving to retire Picasa Web Albums entirely. First, they made it so that all of your Picasa albums are available via Google+ Photos, then they've set Picasa 3 desktop application's sharing options to Google+ as the default, then they even started redirecting https://picasaweb.google.com/home address to Google+ Photo Albums, and Android Gallery Sync options are indeed gone in 4.2.2 version. As of this writing, the latest gapps package includes the following change log:
The last line means that the Gallery, which is no longer a stand-alone app, and instead integrated with the Camera app, is an older version in order to preserve the Picasa Sync functionality, while keeping the newer Camera features such as Pohtosphere. Unfortunately this is only available on very few devices such as Nexus 4 (code name "mako"), Nexus 10 (code name "manta"), and the Galaxy Nexus family (code name "tuna"). This is due to the Camera apps not being open source on either Google's Android core apps, or every other manufacturer's UI overlay. It is therefore impossible to get the Picasa sync code working on other devices.
If you are willing to experiment a bit, there is an older version of an add-on to the main gapps package here which may help you get Picasa Sync back. You should flash this