A CPU would be handy, but as a comparison, I have Minecraft on a laptop that is AMD tri-core 2.1 GHz with an onboard ATI 4200, and with optifine, I can get some great performance out of the standard game (no texture pack larger than 16x16). Your current CPU might be alright for your gaming needs--my nieces mostly play Minecraft on my laptop and I haven't noticed much lag.
Again, using optifine correctly, the RAM usage of the game should decrease. I heard that a while ago Minecraft updated so that even in singleplayer it acts as if its in an online match, which introduced some weird latency lag with enemies. If this one-person LAN is real, I could see a little extra RAM being used compared to older versions of Minecraft...
You could definitely get all these for under $350, but making sure your CPU and your RAM works with your motherboard can be a little crappy, especially if it's for older RAM or processors that aren't really manufactured as much. These older pieces can sometimes be marked up in price just to gouge those older users who can't afford the newer and cheaper stuff.