If you cannot launch Rust in the new version, I would close all extra tabs (Skype, TeamSpeak (which should be fine if you are in a low quality voice chat room, which uses less bandwidth), RaidCall, things that use alot of bandwidth), and launch it at poorest graphics.
Becareful of the ping of the server you are on. The Official server has lots of players and buildings, so that could be affecting your ping. Join Community or Modded servers with not that many people and see if your ping goes down.
If that doesn't work, you might have to play what Garry classifies the "Old Version" of Rust. You can access it with the launch options before starting the game. Still a good game, it just doesn't have awesome graphics and an improved interface like the new version.
I would upgrade your rig. I personally need to upgrade my GeForce GTX 660 Ti graphics card.