Self-repairing, if the HDMI port is burned out, would imply disassembling the PS3 and actually checking where the issue is; This will VOID the warranty, if there's still one. Mainly, it will mean a minimum amount of soldering will be needed, but if it's just part of the embedded circuit, a wired bypass should be more than enough, and an easy task with a steady hand; if it's the port itself, however, you need to de-solder the port first, and then change it with a new one, and that could be slightly more complicate.
If you're not that handy with DIY electronics or if the warranty is still valid, I would take it to a repair shop that specializes in electronics, such as PCs, or send it back to Sony's technical support; possibly look for privates tho, as big chains will likely refuse you unless you have a valid warranty, and if you're out of warranty Sony will likely make you pay almost as much as a new console...
I say try with the analog cable first and see how it goes. Most games from the PS3 run at 720p anyway, so it's not like you're losing much without HDMI, besides upscaled graphics which rarely look good 😉
If you're fine with how analog looks, keep running on it; if you don't, repair it.