Reddit reviews 3rd Party AC Power Adapter (Atari 2600)

It's far more than audio-only - it also carries power and can send/receive data. Examples of what 3.5mm jacks can be used for:

• Reading credit cards
  
  
• Charging and sending data to/from MP3 players
  
  
• Transmitting IR signals
  
  
• Sending CV/gate signals
  
  
• Plugging in computer keyboards
  
  
• Sending shutter release signals to cameras from a distance
  
  
• Powering video game consoles
  
  And surely tons of other things.

You definitely don't want to plug that power adapter into the wall outlet unless you're certain that it's both the right voltage and the right polarity. The best way to ensure this, of course, is to buy another Atari 2600 power supply online, but if you google around, you can probably find out what voltage and polarity you will need. This one on Amazon looks like the one you need from a quick glance, so it might be a good starting point for your search.

As for hooking it up to your TV, here is a great video showing you all you will need to know. If you're ambitious and you trust your skills with a screwdriver, you might want to open up that console and clean it and restore it. It must have decades of accumulated dust and dirt inside it, and it looks like the console end of the RF cable needs to be reseated too.

You should also clean the connectors of all those game cartridges before you use them, and here is one method of doing that.