For that budget your best performing laptop for gaming would be one with an AMD A8 or A10 apu equipped. The integrated graphics chip is no power house like a dedicated one but they are cheap and can do some good gaming, unlike their intel hd 4000 counterparts which in general only allow gaming on the lowest of settings.
IMHO, for $350 you would be better off buying a gaming console like an XBOX One or PS4. If you buy one used, you can save $100 or more versus buying brand new. If you must have a gaming laptop due to transportability/travel requirements, then your best bet is to buy a used AMD-based laptop as suggested by bootcher.
FYI, minimum system requirements to play Archeage:
CPU: Intel Core 2 Duo E4400 or AMD Athlon 64 X2
RAM: 2 GB
OS: Windows Vista
Video Card: NVIDIA GeForce 8400 (512MB VRAM) or ATI Radeon HD 4250 (512 MB VRAM) or Intel HD Graphics 3000
Sound Card: Yes
Free Disk Space: 30 GB
DirectX: 10 (Windows Vista, Windows 7, Windows 8, Windows 8.1)