Well a blank screen is often caused by a dodgy graphics card but you say that sometimes it shows the Windows logo and then switches off so all the evidence so far points to something wrong with your power supply.

Fortunately, if you're happy about changing it yourself (looks like it will be pretty easy), then it's not too expensive, so it won't be a disaster if it turns out that that doesn't fix it.

In fact, I've just had a look on the US web site for Insight (whose UK division are a good supplier over here) and they've got 250 watt PSUs (which will probably be adequate for your older system) for $21 plus shipping.