Cannot access Plex Server on local network

Read these documents:

Installation & Setup of Plex Media Server on Synology

Synology FAQ - Questions, Answers, and a few How-To’s

For initial setup, using an incognito/private window, point your browser directly at the Plex server.

http://use_nas_real_ip_address:32400/web

Also, make sure you use the current version available at plex.tv. The version in the Synology Package Center is approximately 2 years out of date. The above documents provide instructions on downloading the current version and performing a manual installation.