You probably could find a computer with a 64-bit processor for about 350 bucks. It will not be a machine that is future oriented, but it will be sufficient for quite some time.

A new processor of the most current families is about the price of what you want to/can spend.

For the machine of yours four GB of memory is the best you can do and those RAMs are rather cheap nowadays. (Depending on your point of view it is a worthwile investment or a waste of monetary ressources.) Then save some money to buy a current model later. Some software starts to be issued in 64bit versions only. I won't be long before you can not update software that you already have.


Desktop; i7-2600k, 8 GB mem., Win 10 Pro, BIAB 2017; RB 2017 - latest build
Laptop: i5-2410M, 4 GB mem, Win 10 Pro, BIAB 2017; RB 2017 - latest build