It looks like this was originally intended for telephone service distribution.
There's a few ways to go about it depending on how neat and tidy you want the job to look.
- You can pull those blue wires free from the termination block, crimp them with rj-45 heads, and plug them into your switch. Quicker, Easier, Cheapest.
- You can replace the punchdown block in the picture with a mini patch panel. More work, cost, but nicer finished look. You will then need to also buy a handfull of 6 inch patch cables to connect it up from your switch.
You should trace and label those cables as well so you know where they go.
It's likely that one does not end in your apartment, but is going to the street or a patch panel elsewhere intended to provide phone service. To prevent damage, do not connect this one to your network equipment.
(Lots of) Home runs are good
You are correct that you want to run a cable to each room from the central switch. In fact, I would run at least 1 more cable than you think you will need to each room, and consider running a line or 2 to other rooms as well - especially if your walls are open. Cable is cheap, and pulling 4 cables instead of 3 is no more work when done at the same time. If you decide later that you want a 4th jack, you either need a small switch (which does limit bandwidth, not really increase latency) or you need to open walls again to pull that 4th cable.
Use a patch panel
Rather than run the cable from the big switch to each room, you should have a patch panel in between. Patch panels basically change the type of connection on the cable (the back is a 110 punch down block, front is an RJ-45 jack), and are a simple pass-through.
This is to ease installation. Pulling cable through walls is best done when the cable is un-terminated. Terminating the cable (i.e., putting the RJ-45 jacks on the end) can and is done, but punching the cable down into a patch panel is so much easier, especially for someone who has never done it before (and it sounds like neither you nor your electrician has). The cost is marginal (again, go bigger than you think you need now), but you save on headaches during installation.
You would then get keystone jacks that allow you to punch down the cable on the other end:
You shove these into wallplates on an electrical box or low-voltage plate:
They make wall-plates with different numbers of openings (usually 1-6), so you can get what you need for each room.
Finally, you would need short (1-2 ft) "patch" cables to connect the patch panel to the big switch. Buy these cables pre-made, as you won't be able to make your own for less. These are typically stranded cable, as it's more flexible.
Your final setup would look something like this:
(the top-most device with jacks is the patch panel, the middle on is the switch, and the bottom would be your router)
Buy solid copper UTP (unshielded twisted pair) cat5e or cat6 cable, rated properly (usually CMR for typical in-wall installation, but you'll need Plenum if you plan to run it in HVAC ducts), and buy multiple boxes if possible. Standard is 1000 ft but smaller lengths are available, and they come in all different colors. A decent-sized house could take 2000-3000 ft of cabling or more, depending on how many runs and where the network closet is. Again, the more boxes you have, the easier installation will be (you typically pull 1 from each box at the same time, so if you want 4 runs to a single location, having 4 boxes is easiest).
If you want things a little cleaner, you can get a wall-mounted mini rack as well:
Just make sure to get one that has the depth and vertical space (measured in "U") you need. They also make ones with hinges that make patch panel installation a bit easier.
Most product images taken from monoprice.com
Best Answer
Stated another way, your options are to make the best use possible of the cable that's already installed or install new cable that suits your need better.
Ethernet outdoors
There exist environmentally hardened Ethernet switches designed for outdoor use, but you won't find them on the shelf at the local big box electronics store and they'll cost more than consumer/SOHO indoor-rated gear. Then again, indoor-rated gear might perform acceptably for you even outdoors. So there's one option: install an Ethernet switch near enough to all those cables outdoors so that they can be patched into the switch. You'd have to get power to it from somewhere and it would require some kind of protection from water.
Turn cables indoors
Think about what's on the interior side of the wall from that outdoor junction box. Would it be acceptable to place a patch panel near that location? You could cut an opening in the interior wall, pull all the cables from the junction box into a new box inside, and repair the wall.
Modify, abandon, or replace
If all else fails you'll have to install some new cable. Some of the existing cable might run through accessible areas like the attic. Any existing cable you can find and re-route to the new patch panel location saves work in fishing new wire through walls. Consider whether it's useful to have more than one patch (and switch) location to so that everything doesn't have to home-run to a single place.
Don't go to great pains to rip out wire you're not using -- if it isn't hurting anything, just leave it in place. Maybe you or a future owner will think of something to use it for later.
Retrofitting cable is something of an art form, but it can be learned. Things to consider as you plan a retrofit are which direction the floor joists go and where there are mechanical chases (ie for duct work) that you might be able to push wire through.