Let's start by getting the new version of Plex installed. The latest version is 0.9.12.8 and you can find it on the downloads page by clicking on Plex Pass or Public then click on NAS. Make sure Synology is selected and then select the package that is appropriate for your NAS, which is a ??????
Note: Do not use Chrome or Safari to download the package, they have been known to not complete the download correctly.
Then open package center->settings and set the trust level to allow 'any publisher'. Next, click on Manual Install. Then click on browse and navigate to where ever you saved the downloaded package, select it and click open.
The rest is easy, just follow the on screen prompts. When it's done, you may need to start Plex, in which case go to package center->installed->Plex and click on the 'Action' drop down and select run. It will tell you if Plex is running or stopped under 'Status'.
Then click on the DSM start menu (top left of screen) and click on the Plex icon. Plex should then open in your browser.
Installing Plex like this will NOT effect your libraries, assuming you have already setup Plex. If not, then go ahead and create your libraries.
You should NOT store your media in the default 'video' folder on the NAS, if you have then open DSM control panel->shared folders and create a folder named Media or similar and make sure it has read/write permissions for both Plex and admin. Then move all of your media into the new folder you just created.
If you already have libraries setup, then edit those libraries in Plex by clicking on the pencil icon beside the library name and change the folder address to the new location. Plex should not need to download new metadata, so you should not have any problems, just make sure you run an update on the library after changing the folder path.
Let me know if that has solved your issue or not.