I'm afraid you will be disappointed with the Radeon HD 6310's performance. If you click the following link you can find some game benchmarks for that card. Overall, it is slower than the Intel HD 3000 graphics core that is inside a Core i3 2310m CPU. As a comparison, the Intel HD 3000 graphics core is a little better than the desktop version of the Radeon HD 5450.
These are the minimum system requirements for Star Trek Online. I'll also add that these are specs for a desktop PC. Laptops are generally less powerful than their desktop counterparts.
OS: Windows XP SP2 / Windows Vista / Windows 7 (32 or 64-bit)
CPU: Intel Core 2 Duo 1.8 Ghz or AMD Athlon X2 3800+
Memory: 1GB RAM
Video: NVIDIA GeForce 7950 / ATI Radeon X1800 / Intel HD Graphics
Sound: DirectX 9.0c Compatible Soundcard
DirectX: Version 9.0c or Higher
HDD: 8GB Free Disk Space
Network: Internet Broadband Connection Required
Disc: 6X DVD-ROM
The AMD E-350 is 1.6GHz, just slightly below the minimum of 1.8GHz. Not too bad, but I don't know how the AMD E-Series CPU compares to an AMD Athlon X2.
The major problem would be the Radeon HD 6310. Sure it is technically 5 generations newer than the Radeon X1800, but the Radeon X1800 was a powerful card compared to the wimpy Radeon HD 6310.
You need to spend more than $330 on a laptop if you want to play games. At least double that, at the minimum just to get decent performance for casual games; not fairly demanding games (for laptops) like Star Trek Online.