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). Movies can be stored on its internal HDD. If you need more storage, you can swap the HDD for a larger newer model, or plug in an external USB HDD.
Edit: To copy movies to it over the network, you'll need to enable network sharing on your plex server. If it's running Windows, just share the Plex media folders with an account and password of your choosing. if it's running Linux, you'll need to install and configure Samba - that's the package which allows Windows network shares. (I mention Linux because my Plex server runs on a Linux virtual machine with 1 GB of RAM and typically only uses about 500 MB.)