As stated above, the Dell Dimension 4500 used the old (notorious) P4 series of processors. It will probably burn around 100 Watts at idle. If you pay the U.S. average electricity rate of 12 cents/kWh, then every Watt used by a device left on 24/7 will cost you about $1 per year. So your 100 Watt Plex server will cost you about $100/yr to leave on.
Just buy a used Core i laptop off eBay, maybe even one with a broken screen or keyboard (as long as you can use an external monitor and keyboard to configure it), preferably with an Ethernet port. Those typically only use 3-8 Watts at idle. And anything that's i3 or better should have enough horsepower to transcode 1080p video (I'm not even sure the P4 is powerful enough to transcode)...