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Possibly SSD if you have some more money to throw out. Anyway, as he said, this is by no means setup to do serious gaming.

+1

+1

My GS3 outperforms that thing for sure.

this probably!

That thing won't even handle 64bit applications on that ancient CPU, so web browsing is likely off the table.

OP should, at worst, get a dual core. Quad is preferable still though.

You essentially cannot build a computer that will do even the majority of baseline gaming for that little.
To do any reasonable gaming, you're looking at nearly $200 for just a decent mid-range GPU.
To build a computer with gaming in mind, even just at the average minimum specs, you're going to need a budget more like $500-600 minimum.

The breakdown is about like this:
$100 CPU
$80 MoBo
$80 RAM
$190 GPU (R9 270x, can go a bit lower, like R7 260, but this is ideal for mid-range)
~$100 PSU

Those figures, except the GPU, are just rough estimates based on some averages in mind because how often I do this. They are by no means firm figures with specific parts in mind, just something approximating the baseline for what I'd call decent components. There is some wiggle room in there of course, but the purpose is to give an idea of where a reasonable baseline budget target should be.

So, basically, to get something remotely reasonable assembled, you are going to have to roughly double your budget.
That doesn't even include storage or optical drives either yet, but some things can be adjusted to account for them.
If you also need a new monitor and other peripherals, then you're looking to probably be pushing $750+.

The price range you're at is simply unreasonable for a new build. Effectively your options are this: Search classified ads/craigslist for a good deal on a used gaming rig (which I never recommend willingly), or save up quite a bit larger total first.

To be totally blunt: Building anything in that price range is a complete waste of money. It will be an extremely low end PC, probably with very cheap and unreliable parts, and liable not only to be useless as soon as it is built, but to be prone to having a very short lifespan. My advice as a PC builder and general tech geek, since I was 13-14 (that's 1996 for anyone counting, meaning yes, I have been doing this almost 20 years counting non-professional years), is that the baseline I give is a baseline for reasonable return on investment. ie: $250 for a PC that lasts ~1 year as a terrible quality experience, or $600 and I can build you a PC you can game on for at least 3 years of a decent quality experience, double the price for 3x the life and 10x the experience.
As such, unless absolutely forced by a customer, I will never even attempt to build a setup at such a low price point.

Also, where you say your current PC is 7-10 years old, I'll simply tell you outright right now, you will not be salvaging parts from it reasonably. Anything in it is going to be effectively useless, and it's so far past the reasonable life expectancy of PC parts, I wouldn't risk putting them into a new machine. Some of the very longest warranties in the business top out about 5 years for a reason. If you've had stuff continue working to the 10 year mark, you've already doubled that. It's not unheard of, but denotes quality parts as well as luck. The saying "don't press your luck" comes to mind regarding trying to reuse such components.

I don't think you're likely to find a sub-$350 rig which will play the majority of games on the default settings (well, technically may be yes, the majority of games are probably small indie or old, but I don't imagine that's what you mean.)

The combo in the original link looks to me like pretty good value, but you need Windows and a decent graphics card to get a real gaming system, and each of these cost around $100.

That said, if you have a copy of Windows you can transfer from your previous PC, that may be enough.